text:
\section{Introduction}
In conformal geometry, one of the most fundamental problems is understanding the relationship between conformally covariant operators, their associated conformal invariants and the related PDEs.

As a first example, let us consider the Laplace-Beltrami operator in two dimensions on a closed surface $(M,g)$ and the  Gaussian curvature.  Through a conformal change of metric $g_v=e^{2v}g$, we have the associated PDE
\begin{equation}
\label{1.curvature-q}
-\Delta_gv+K_g=K_{g_v}e^{2v},
\end{equation}
where $\Delta_g$ denotes the Laplace-Beltrami operator with respect to the background metric $g$, $K_g$ and $K_{g_v}$ are the Gaussian curvatures of the metric $g$ and $g_v$, respectively. Observe that the latter equation yields in particular the conformal invariance of the total Gaussian curvature which is then tight to the topology of the surface via the Gauss-Bonnet formula
$$
\int_M K_g\,dvol_g=\chi(M).
$$
Here, $\chi(M)$ is the Euler characteristic of the surface.

A classical issue here is the prescribed Gaussian curvature problem or the Uniformization Theorem about the existence of a conformal metric in the conformal class of $g$ with prescribed (possibly constant) curvature. This amounts to solve the PDE in \eqref{1.curvature-q} which has been systematically studied since the works of Berger \cite{berger}, Kazdan-Warner \cite{kw} and Chang-Yang \cite{cy1,cy2}.

\

In higher dimensions, we have the so-called GJMS operators $P_g^{2m}$ and the related $Q$-curvatures $Q_g^{2m}$ which are the higher-order analogues of the Laplace-Beltrami operator and the Gaussian curvature for $2m$-dimensional closed manifolds, see \cite{gjms,GZ}. These are conformally covariant differential operators whose leading term is $(-\Delta_g)^m$.
In particular, when $m=1$ we recover the Laplace-Beltrami operator and the Gaussian curvature. Moreover, for $m=2$, $P_g^{4}$ and $Q_g^4$ are related to the Paneitz operator and the standard $Q$-curvature:
\begin{equation}
\label{1.paneitz}
\begin{aligned}
P_g^4f&=P_g f=\Delta_g^2f+\mbox{div}_g\left(\frac23R_gg-2\mbox{Ric}_g\right)df,\\
Q_g^4&=2Q_g=-\frac1{6}(\Delta_gR_g-R_g^2+3|\mbox{Ric}_g|^2),
\end{aligned}
\end{equation}
where $\mbox{Ric}_g$ and $R_g$ stands for the Ricci tensor and the scalar curvature of the manifold $(M,g)$. See the original works of Paneitz \cite{pan1,pan2} and Branson \cite{branson} for more details.

\

The family of GJMS operators and the related $Q$-curvature functions play now an important role in modern differential geometry. As in the lower order case, under the conformal transformation $g_v=e^{2v}g$, $P_g^{2m}$ and $Q_g^{2m}$ satisfy the following law
\begin{equation}
\label{1.higher}
P_{g_v}^{2m}=e^{-2mv}P_g^{2m},\quad P_g^{2m}v+Q^{2m}_g=Q_{g_v}^{2m}e^{2mv}.
\end{equation}
Again, the total $Q$-curvature $\int_M Q^{2m}_g \,dvol_g$ is seen to be conformally invariant for which the Chern-Gauss-Bonnet formula holds true. In analogy with the two-dimensional case, a core problem is the prescribed $Q$-curvature problem which is in turn related to the solvability of \eqref{1.higher}.

\medskip

One can attack this problem variationally by looking at the critical points of the associated energy functional. A lot of work has been done in this direction, in particular for the four-dimensional case and the Paneitz operator \eqref{1.paneitz}. In this setting, assuming
$$
P_g\geq0, \quad \mbox{Ker}\{P_g\}=\{constants\},
$$
the problem has been first solved by Chang-Yang \cite{cy3} for
$$
\int_M Q^{4}_g \,dvol_g=2\int_M Q_g \,dvol_g<16\pi^2=2\int_{\mathbb{S}^4}Q_{g_0}\,dvol_{g_0}.
$$
Here, $g_0$ is the standard metric of the sphere. See also the related work of Gursky \cite{gursky}. This is the so-called subcritical case in which the energy functional is coercive and bounded from below by means of the Adams-Trudinger-Moser inequality \cite{adams} and solutions corresponds to global minima using the direct methods of the calculus of variations.  We refer to the discussion in the sequel for the precise definition of the subcritical, critical and supercritical case. The supercritical case $\int_M Q^{4}_g \,dvol_g>16\pi^2$, where the energy functional fails to be bounded from below, has been considered by Djadli-Malchiodi \cite{dm} via a new min-max method based on improved versions of the Adams-Trudinger-Moser inequality, solving the problem provided
$$
	\mbox{Ker}\{P_g\}=\{constants\}, \quad \int_M Q^{4}_g \,dvol_g \notin 16\pi^2\mathbb{N}.
$$
Finally, some existence results for the critical case $\int_M Q^{4}_g \,dvol_g \in 16\pi^2\mathbb{N}$ have been derived by Ndiaye \cite{ndiaye2} by making use of the critical point theory at infinity jointly with a blow-up analysis.

\medskip

As far as the higher-dimensional case $2m>4$ is concerned, the subcritical case has been solved in \cite{brendle} via a geometric
flow, while the Djadli-Malchiodi's argument has been generalized by Ndiaye \cite{ndiaye} to treat the supercritical case.

\

In this paper we are interested in prescribing the $Q$-curvature on a general $2m$-dimensional closed manifold $M$ with conical singularities. Let $g$ be a smooth metric on $M$. We will say that a point $q\in M$ is a conical singularity of order $\alpha\in(-1,+\infty)$ for the new metric $g_v=e^{2v}g$ if
$$
	g_v(x)=f(x)|x|^{2\alpha}|dx|^2 \quad \mbox{locally around } q,
$$
for some smooth function $f$. The set of conical singularities $q_j$ of orders $\alpha_j$ is encoded in the divisor
$$
	D=\sum_{j=1}^N \alpha_j q_j,
$$
while $(M,D)$ will denote the related conical manifold. We define
$$
	\kappa_{g}=\int_MQ_{g}^{2m}\,dvol_{g}, \quad \kappa_{g_v}=\int_MQ_{g_v}^{2m}\,dvol_{g_v},
$$
for which the following relation holds
\begin{equation} \label{kappa}
\kappa_{g_v}=\kappa_{g}+\dfrac{\Lambda_m}{2} \sum_{j=1}^N \alpha_j,
\end{equation}
where $\Lambda_m=(2m-1)!|\mathbb{S}^{2m}|$. This can be regarded as a singular Chern-Gauss-Bonnet formula, see for example Theorem~\ref{CGB}. The critical threshold of a singular manifold is essentially related to the singular Adams-Trudinger-Moser inequality stated in Theorem \ref{ATM}. In the spirit of Troyanov \cite{troy} we let
$$
\t(M,D)=\Lambda_m\Bigr(1+\min_j\left\{\a_j,0\right\}\Bigr)
$$
and give the following classification.

\begin{definition} The singular manifold $(M,D)$ is said to be:
$$\begin{array}{ll}
\hbox{subcritical}&\hbox{if} \quad \kappa_{g_v}<\t(M,D)\\
\hbox{critical}&\hbox{if} \quad \kappa_{g_v}=\t(M,D)\\
\hbox{supercritical}&\hbox{if} \quad \kappa_{g_v}>\t(M,D).
\end{array}$$
\end{definition}
See also the recent work of Fang-Ma \cite{FangMa} for a similar discussion. We point out we have a slightly different notation for $\Lambda_m$ with respect to the latter paper.

\

Due to the singular behavior of the conformal factor $v$ around a conical point, prescribing the $Q$-curvature on a manifold with conical singularities at $q_j\in M$ of order $\alpha_j\in(-1,+\infty)$ is related to the solvability of the following singular PDE
\begin{equation}
\label{1.s-mliouville}
P_g^{2m}v+Q_g^{2m}=Q_{g_v}^{2m}e^{2mv}-\frac{\Lambda_m}{2}\sum_{j=1}^N\alpha_j\delta_{q_j},
\end{equation}
where $\delta_{q_j}$ stands for the Dirac measure located at the point $q_j\in M$. One may desingularize the behavior of $v$ around the conical points by considering
\begin{equation*}
u=v-\frac{\Lambda_m}{2}\sum_{j=1}^N\alpha_jG(x,q_j),
\end{equation*}
where $G(x,p)$ is the Green function of $P_g^{2m}$, see for example Lemma \ref{expGreen}. Then $u$ satisfies
\begin{equation}
\label{1.mliouville}
P_g^{2m}u+Q_g^{2m}+\dfrac{\Lambda_m}{2|M|} \sum_{j=1}^N \alpha_j=\widetilde Q e^{2mu},
\end{equation}
where
\begin{equation} \label{qtilde}
\widetilde Q=Q_{g_v}^{2m}e^{-m\Lambda_m\sum_{j=1}^N\alpha_jG(x,q_j)},
\end{equation}
which is now singular at the points $q_j$.

\medskip

The singular equation \eqref{1.mliouville} has been studied mainly in the two-dimensional case, that is in relation to the prescribed Gaussian curvature problem. After the initial work of Troyanov \cite{troy}, there have been contributions by many authors, as for example \cite{ChenLi1,ChenLi2,ChenLi3,LuoTian,McOwen}. This problem has received a lot of attention also in recent years, see \cite{bdm,bm,bt,cm,mr}. See also \cite{eremenko,MazzeoZhu,MondPanov1,MondPanov2} for further developments in this direction.

\medskip

In the higher-dimensional case $m>1$ there are very few results available. The subcritical regime have been just recently solved by Fang-Ma \cite{FangMa}, where the four-dimensional case is considered. The authors point out their method could be applied for higher dimensions too. In any case, the existence here follows by direct methods of the calculus of variations once the singular Adams-Trudinger-Moser inequality in Theorem \ref{ATM} is derived. See also \cite{hlw} for a related result on the sphere via a fixed point argument. For a blow-up analysis in dimension four we refer instead to \cite{AhmedouWuZhang}. Concerning the existence problem in the supercritical case, the only result we are aware of is \cite{hmm} where the authors consider a slightly supercritical problem on the sphere, again with a fixed point argument in the spirit of \cite{hlw}.

\medskip

The goal of this paper is to give a first general existence result for $2m$-dimensional conic manifolds in the supercritical regime. We define a critical set of values $\Gamma$ as follows:
\begin{equation} \label{gamma}
\Gamma=\left\{n\Lambda_m+\Lambda_m\sum_{i\in J}(1+\alpha_i)\mid n\in\mathbb{N}\cup\{0\}\quad\mathrm{and}\quad
J\subset\{1,\dots,N\}\right\}.
\end{equation}
Observe that if $\alpha_j\in\mathbb{N}$ for all $j$, then we simply have $\Gamma=\Lambda_m\mathbb{N}$. Recall now the definition of the total singular curvature $\kappa_{g_v}$ given in \eqref{kappa}. Let $M^{\mbox{\tiny{R}}}\subset M$ be a closed $n$-dimensional submanifold, $n\in[1,2m]$, such that $\a_j\notinM^{\mbox{\tiny{R}}}$ for all $j=1,\dots,N$. Then, we have:
\begin{theorem}
\label{th1.2}
Let $(M,D)$ be a supercritical singular $2m$-dimensional closed manifold with $\alpha_j>0$ for $j=1,\dots,N$ such that there exists a retraction $R:M\to M^{\mbox{\tiny{R}}}$. Let $Q$ be a smooth positive function on $M$. If
$$
\mbox{Ker}\{P_g^{2m}\}=\{constants\}, \quad \kappa_{g_v}\notin\Gamma,
$$
then there exists a conformal metric on $(M,D)$ with $Q^{2m}$-curvature equal to $Q$.
\end{theorem}

\smallskip

\begin{remark}\label{rem-ret}
\emph{We point out that a retraction $R:M\to M^{\mbox{\tiny{R}}}$ as above exists for a wide class of manifolds. For example we can consider manifolds of the type $M^{n}\times M^{2m-n}$, where we denote by $M^l$ any $l$-dimensional closed manifold. Indeed, it is easy to see that we can define a retraction $R:M^{n}\times M^{2m-n}\to M^{n}\times\{p\}$ for some $p\in M^{2m-n}$ with the desired properties. Observe that the torus $\mathbb{T}^{2m}$ belongs to this class of manifolds. One could also consider the connected sum $(M^{n}\times M^{2m-n})\# N^{2m}$, modifying the above retraction so that it is constant on $N^{2m}$.}
\end{remark}

\medskip

We can even deduce the following multiplicity result. Here, $M^{\mbox{\tiny{R}}}_k$ are the formal barycenters of $M^{\mbox{\tiny{R}}}$ according to \eqref{bary} and $\widetilde H_q(M^{\mbox{\tiny{R}}}_k)$ denotes its reduced $q$-th homology group.
\begin{theorem} \label{th-mult}
Under the assumptions of Theorem \ref{th1.2}, let $\kappa_{g_v}\in(k\Lambda_m,(k+1)\Lambda_m)$. Then, if  $\mathcal E$ in \eqref{functional} is a Morse functional,
$$
	\#\{ \mbox{solutions of }\eqref{1.mliouville}\} \geq \sum_{q\geq0} \mbox{\emph{dim} } \widetilde H_q(M^{\mbox{\tiny{R}}}_k).
$$
\end{theorem}

\begin{remark}
\emph{Consider for example the class of manifolds $M^{n}\times M^{2m-n}$ in Remark \ref{rem-ret}. We will get an explicit lower bound on the number of solutions as far as we can explicitly estimate the homology groups of $M^{n}_k$. One can find such computations in \cite{de} for general manifolds $M^n$, focusing on the cases $n=2$ and $n=4$. For some simple manifolds we can easily compute the homology groups. For example, if $M^{n}$ is a $2$-dimensional $G$-torus (connected sum of $G$ tori), then we have at least $\frac{(N+G-1)!}{N!(G-1)!}$ solutions, see \cite{bdm}.}
\end{remark}

\

The argument of the proof of the existence result is in the spirit of the celebrated min-max scheme of \cite{dm}, extended to high dimensions by \cite{ndiaye}, jointly with some ideas of \cite{bdm} to treat the singularities. Roughly speaking, the strategy is based on the study of the sublevels of the energy functional, in particular by showing the low sublevels are non-contractible. This is done by using improved versions of the singular Adams-Trudinger-Moser inequality. We will then overcome the complexity due to the singularities by retracting the manifold onto $M^{\mbox{\tiny{R}}}$, not containing the singular points. This leads us to study the low sublevels just by looking at functions concentrating on such submanifold which is enough to gain some non-trivial homology.

\medskip

To conclude the min-max argument we would need some compactness property as the Palais-Smale conditions is not available in this setting. We thus use Struwe's monotonicity trick \cite{struwe}, which is by now a standard tool in this class of problems, to deduce the existence of a sequence of solutions $u_k$ satisfying \eqref{1.mliouville}. We will then conclude by showing the following compactness result which actually holds for any $2m$-dimensional manifold and $\alpha_j>-1$.
\begin{theorem}
\label{th1.1}
Let $u_k$ be a sequence of solutions of \eqref{1.mliouville} with $\widetilde Q>0$ and $\alpha_j>-1$ for $j=1,\dots,N$. If
$$
\mbox{Ker}\{P_g^{2m}\}=\{constants\}, \quad \kappa_{g_v}\notin\Gamma,
$$
then there exists a constant $C$ independent of $k$ such that
\begin{equation*}
\|u_k\|_{L^\infty(M)}\leq C.
\end{equation*}
\end{theorem}

The latter result is a consequence of a quantization phenomenon of blowing-up solutions which is derived via Pohozaev-type inequalities in the spirit of \cite{AhmedouWuZhang,bt,LinWei}.

\medskip

The above analysis, together with Morse inequalities, allows us to deduce also the multiplicity result of Theorem \ref{th-mult}.

\begin{remark}
\emph{We conclude the introduction with the following observations.}

\medskip

\noindent 1. \emph{The existence result is derived for the case $\alpha_j>0$ for all $j$. In principle, the same strategy can be carried out for the case $\alpha_j\in(-1,0)$. However, in this scenario we get a worse Adams-Trudinger-Moser inequality in Theorem \ref{ATM} and this in turn affects the topology of the low sublevels in a non-trivial way, see for instance \cite{cm}. We postpone this study to a future paper.}

\medskip

\noindent 2. \emph{The same analysis should work in the odd-dimensional case with some further technical difficulties, as explained in Section 5 of \cite{ndiaye}. We will not discuss this case in the present paper.}
\end{remark}

\medskip

This paper is organized as follows: in Section 2 we collect some useful preliminary results, Section 3 is devoted to blow-up analyis and the proof of Theorem~\ref{th1.1} and in Section 4 we carry out the min-max method to derive the existence and multiplicity results of Theorems \ref{th1.2} and \ref{th-mult}. A Pohozaev-type identity is provided in the last section.

\medskip

\begin{center}
\textbf{Notations:}
\end{center}
\begin{enumerate}
\item [$B_r^{M}(p)$] \quad the ball centered at $p\in M$ with geodesic radius $r$ on the manifold $M$.
\smallskip
\item [$B_r(p)$] \quad the ball centered at $p$ with radius $r$ in $\mathbb{R}^{2m}$.
\end{enumerate}

\

\section{Preliminary facts}

In this section we recall briefly some known results which can be easily derived from the existing literature.  Let $p$ be a point in $M$ and $B_r^M(p)$ be a neighborhood of $p$ such that $B_r^M(p)$ is mapped by $\exp_p^{-1}$ diffeomorphically onto a neighborhood of $0\in T_p(M),$ where $T_p(M)$ refers to the tangent space of $p$ which can be identified with $\mathbb{R}^{2m}$. The local coordinates defined by the chart $(\exp_p^{-1},B_r^{M}(p))$ are called normal coordinates with center $p.$ In such coordinates, the Riemannian metric at the point $p$ satisfies
\begin{equation}
\label{a.m1}
g_{ij}=\delta_{ij},~g_{ij,k}=0,~\Gamma_{jk}^i=0,\quad \mbox{for all}~i,j,k\in\{1,\cdots,2m\},
\end{equation}
where $\Gamma_{jk}^i$ stands for the Christoffel symbols. With the above preparation, we have
\begin{lemma}
\label{lea.m}
Let $-\Delta_g$ be the Laplace-Beltrami operator and $p$ be any point of $M$. In normal coordinates at $p$, we have
\begin{equation}
\label{2.expand}
(-\Delta_g)^mu=(-\Delta)^mu+\mathcal{D}^{2m}u+\mathcal{D}^{2m-1}u,
\end{equation}
where $\mathcal{D}^{2m}$ is a linear differential operator of order $2m$ whose the coefficients are $O(|x-p|^2)$ as $x$ tends to $p$, while $\mathcal{D}^{2m-1}$ is a linear differential operator of order at most $2m-1$, and whose coefficients  belong to $C_{\mathrm{loc}}^l(\mathbb{R}^{2m})$ for all $l\geq 0$.
\end{lemma}

\begin{proof}
By the definition of Laplace-Beltrami operator, we have
\begin{equation}
\label{a.m2}
-\Delta_gu=-\frac{1}{\sqrt{\mbox{det}g}}\partial_i(\sqrt{\mbox{det}g}g^{ij}\partial_ju),
\end{equation}
where $g^{ij}$ is the inverse of $g_{ij}.$ Using \eqref{a.m1} we can write
\begin{equation}
\label{a.m3}
-\Delta_gu=-g^{ij}\partial_{ij}u-\frac{\partial_i(\sqrt{\mbox{det}g}g^{ij})}{\sqrt{\mbox{det}g}}\partial_ju.
=-g^{ij}\partial_{ij}u-\vartheta_j\partial_ju,
\end{equation}
where $\vartheta_j$ is a smooth function. Based on \eqref{a.m3}, it is easy to see that in the final expression of $(-\Delta)_g^m$ the leading differential order  is
$$g^{i_1j_1}g^{i_2j_2}\cdots g^{i_mj_m}\partial_{i_1j_1i_2j_2\cdots i_mj_m}\cdot,$$
where $i_a,j_b\in\{1,\cdots,2m\},~\forall a,b\in\{1,\cdots,m\}.$
While the left terms are order at $2m-1$ and the coefficients are smooth due to the exponential map is differentiable with arbitrary order. Consider the leading term, using \eqref{a.m1}, we see that
$$g^{ij}(x)=\delta_{ij}(x)+O(r^2),\quad \mathrm{if}~x\in B_r^M(p).$$
Therefore we can write
$$\prod_{a=1}^mg^{i_aj_a}(x)=\prod_{a=1}^m\delta_{i_aj_a}+O(r^2),\quad \mathrm{if}~x\in B_r^M(p).$$
As a consequence, we can write
\begin{equation}
g^{i_1j_1}g^{i_2j_2}\cdots g^{i_mj_m}\partial_{i_1j_1i_2j_2\cdots i_mj_m}\cdot
=(-\Delta)^m\cdot+\mathcal{D}^{2m}\cdot,
\end{equation}
with $\mathcal{D}^{2m}$ satisfies the property stated in the lemma. Then we finish the proof.
\end{proof}

\medskip

\begin{remark} \label{rem-local}
\emph{Throughout the paper, when performing local computations, we may consider conformal normal coordinates, see \cite{cao} or \cite{LeeParker}, if needed. These are normal coordinates at a point $x_0$ for a metric $g_w=e^{2w}g$ with \emph{det}$(g_w)=1$ in a small neighborhood of $x_0$ and other useful properties, for which we refer the interested reader to \cite{WeinZhang}. Observe that the differential operator $P_g^{2m}$, after this change of the metric, can be still expanded as the right hand side of \eqref{2.expand}. Indeed, $w(x)=O(d_{g_w}^2(x,x_0))$ and it is smooth in a small neighborhood of $x_0$. Moreover, by \eqref{1.higher} we can write \eqref{1.mliouville} as
$$
P_{g_w}^{2m}u=e^{-2mw}P_gu=e^{-2mw}(-Q_g^{2m}+\widetilde{Q}e^{2mu}),
$$
which is equivalent to
$$
e^{2mw}P_{g_w}^{2m}u=-Q_g^{2m}+\widetilde Qe^{2mu}.
$$
Concerning the differential operator $e^{2mw}P_{g_w}^{2m}$, it is known that the leading order operator is $e^{2mw}(-\Delta_{g_w})^m$. Then, by using the above Lemma \ref{lea.m} and the asymptotic behavior of $w(x)$, we can write
$$
e^{2mw}(-\Delta_{g_w})^{m}u=(-\Delta)^mu+\mathcal{P}^{2m}u+\mathcal{P}^{2m-1}u,
$$
with $\mathcal{P}^{2m}$ and $\mathcal{P}^{2m-1}$ satisfying the same properties as $\mathcal{D}^{2m}$ and $\mathcal{D}^{2m-1}$ in Lemma \ref{lea.m}.}
\end{remark}

\

In what follows, recall $\Lambda_m=(2m-1)!|\mathbb{S}^{2m}|$. We will need the following structural result on the Green functional of the operators under consideration, see Lemma 2.1 in \cite{ndiaye}.
\begin{lemma}\label{expGreen}
Suppose $\mbox{Ker}\{P_g^{2m}\}=\{constants\}$. Then the Green function $G(x,y)$ of $P_g^{2m}$ exists and has the following properties:
\begin{enumerate}
\item For all $u \in C^{2m}(M)$ we have for $x \neq y \in M$
$$
u(x)-\bar u=\int_M G(x,y)P^{2m}_g u(y)dV_g(y), \quad \int_M G(x,y)P^{2m}_g dV_g(y)=0,
$$
$$P_g^{2m}G(x,p)=\delta_p-\dfrac{1}{|M|},$$
where $\bar u$ is the average of $u$.
\item The function
$$G(x,y)=H(x,y)+K(x,y)
$$
is smooth on $M\times M$, away from the diagonal. The function $K$ extends to a $C^{2,\alpha}$ function on $M \times M$ and $H$ satisfies
$$
H(x,y)=\frac{2}{\Lambda_m} \log\Big ( \frac{1}{r} \Big )\,f(r),
$$
\end{enumerate}
where $r$ is the geodesic distance from $x$ to $y$ and $f$ is a smooth positive, decreasing function such that $f(r)=1$ in a neighborhood of $r=0$ and $f(r)=0$ for $r \geq \text{inj}_g(M)$.
\end{lemma}

As mentioned in the introduction, the total $Q$-curvature is a conformal invariant for which the following singular Chern-Gauss-Bonnet formula holds true.

\begin{theorem}\label{CGB}
Consider the divisor $D=\sum_{i=1}^N p_i \alpha_i$ where $p_i \in M$ and $\alpha_i >-1$. Let $g$ be a smooth metric on $M$ and $g_v=e^{-2mv}g$ be the conical metric representing the divisor as explained before \eqref{kappa}. Then, it holds
\begin{equation}
\label{CGBformula}
\int_MQ_{g_v}^{2m}\,dvol_{g_v}=\int_MQ_{g}^{2m}\,dvol_{g}+\frac{\Lambda_m}{2}\sum_{i=1}^N \alpha_i.
\end{equation}
\end{theorem}
\begin{proof}
The proof is a standard argument (see e.g. \cite{FangMa} for $m=2$), using Lemmata \ref{lea.m} and \ref{expGreen}. See also \cite{BN} for a more general result which implies this statement as a particular case.
\end{proof}

Finally, we state the general singular Adams-Trudinger-Moser inequality suitable to treat our problem. We focus here for simplicity on the case $P_g^{2m}\geq0$ and refer to the discussion in \cite{ndiaye} for the general case.
\begin{theorem}\label{ATM}
Consider the divisor $D=\sum_{i=1}^N p_i \alpha_i$ where $p_i \in M$ and $\alpha_i >-1$. Let $\widetilde Q>0$ be as in \eqref{qtilde}. Assume $P_g^{2m}\geq0$  and   $\mbox{Ker}\{P_g^{2m}\}=\{constants\}$. Then, there exists a constant $C=C(\alpha,M)$ such that for any $u \in H^{m}(M)$ we have
$$
\Lambda_m\Bigr(1+\min_j\left\{\a_j,0\right\}\Bigr)\log \int_M \widetilde Qe^{2m(u-\bar u)}\,dvol_{g} \leq m \int_M uP^{2m}_g u\,dvol_{g} + C,
$$
where $\bar u$ is the average of $u$.
\end{theorem}
\begin{proof}
The case without singularities is Proposition 2.2 in \cite{ndiaye}. The conic case follows by the same approach as in \cite{FangMa}.
\end{proof}

\

\section{Compactness property}
In this section we shall prove the compactness result of Theorem \ref{th1.1}. For simplicity of notation, there is no loss of generality to consider a blow-up sequence $u_k$ to
\begin{equation}
\label{2.u}
P_g^{2m}u_k+Q_g^{2m}= \widetilde Q e^{2mu_k},
\end{equation}
where
$$\widetilde Q=  Q_{g_{v}}^{2m}e^{-m\Lambda_m\sum_{j=1}^N\alpha_jG(x,q_j)}>0, \quad \Lambda_m=(2m-1)!|\mathbb{S}^{2m}|.$$
Theorem \ref{th1.1} will follow by showing a concentration phenomenon:
$$\widetilde Q e^{2mu_k}\rightharpoonup\sum_{p\in\mathcal{B}}
(1+\alpha_p)\Lambda_m\delta_p\quad \mathrm{as}\quad k\to+\infty,$$
weakly in the sense of measures, where $\mathcal{B}$ is the blow up set of $u_k$,
\begin{equation*}
\alpha_p=\begin{cases}
0,\quad &\mbox{if}\quad p\notin\{q_1,\cdots,q_N\},\\
\alpha_j,\quad &\mbox{if}\quad p=q_j.
\end{cases}
\end{equation*}
It follows that when blow-up occurs, then necessarily
$$\int_{M}\widetilde Qe^{2mu_k}\,dvol_g\to \sigma\in\Gamma\quad\mbox{as}\quad k\to +\infty,$$
where $\Gamma$ is given in \eqref{gamma}.

\

First, we establish the following lemma.
\begin{lemma}
\label{le2.1}
Let $\{u_k\}$ be a sequence of functions on $(M,g)$ satisfying \eqref{2.u}. Then for $i=1,\cdots,2m-1$ we have
\begin{equation}
\label{2.est-int}
\int_{B_r^M(x)}|\nabla^i u_k|^l\,dy\leq C(n)r^{2m-il},\quad 1\leq l<\frac{2m}{i},~\forall x\in M,~0<r<r_{inj}.
\end{equation}
where $r_{inj}$ is the injectivity radius of $(M,g).$
\end{lemma}

\begin{proof}
Set $f_k:=\widetilde Qe^{2mu_k}-Q_g^{2m}$, which is bounded in $L^1(M)$. By Green's representation formula we have
\begin{equation}
\label{2.rep-u}
u_k(x)=\Xint-_Mu_k\,dvol_g+\int_MG(x,y)f_k(y)\,dvol_g(y).
\end{equation}
For $x,y\in M,~x\neq y$, we have (see \cite[Lemma2.1]{ndiaye})
\begin{equation}
\label{2.est-g}
|\nabla^i_yG(x,y)|\leq\frac{C}{d_g(x,y)^i},\quad 1\leq i\leq 2m-1.
\end{equation}
Then differentiating \eqref{2.rep-u} and using \eqref{2.est-g} and Jensen's inequality, we get
\begin{equation*}
\begin{aligned}
|\nabla^i u_k(x)|^l\leq~& 	C\left(\int_M\frac{1}{d_g(x,y)^i}|f_k(y)|\,dvol_g\right)^l\\
\leq~&C\int_M\left(\frac{\|f_k\|_{L^1(M)}}{d_g(x,y)^i}\right)^l
\frac{|f_k(y)|}{\|f_k\|_{L^1(M)}}\,dvol_g.
\end{aligned}
\end{equation*}
From Fubini's theorem we conclude that
\begin{equation}
\label{2.estu_k}
\int_{B_r^M(x)}|\nabla^i u_k(x)|^l\,dvol_g\leq C\sup_{y\in M}\int_{B_r^M(x)}\frac{\|f_k\|_{L^1(M)}^l}{d_g(x,z)^{il }}\,dvol_g(z)
\leq Cr^{2m-il}.
\end{equation}
It proves the lemma.
\end{proof}

Next, we shall give the minimal local mass around a blow-up point.

\begin{lemma}
\label{le2.2}
Let the sequence $u_k$ satisfy \eqref{2.u} and blowing-up at $q_j$. Suppose that
\begin{equation*}
\widetilde Qe^{2mu_k}\rightharpoonup\mathfrak{m},~\mbox{weakly in the sense of measure in M},
\end{equation*}
then
$$\mathfrak{m}(q_j)\geq\frac12\min\{\Lambda_{m}(1+\alpha_j),\Lambda_{m}\}.$$
\end{lemma}

\begin{proof}
To show the thesis, it suffices to prove that if the following inequality holds
\begin{equation}
\label{2.bcon}
\int_{B^M(q_j,2r)}\widetilde Qe^{2mu_k}\,dvol_g<\frac12\min\{\Lambda_{m}(1+\alpha_j),\Lambda_{m}\},\quad r<\frac{r_{inj}}{2},
\end{equation}
then
\begin{equation}
\label{2.est}
u_k\leq C~\mathrm{in}~B^M(q_j,r).
\end{equation}
We study equation \eqref{2.u} in terms of the local normal coordinates at $q_j$ (see for example Lemma \ref{lea.m} and Remark \ref{rem-local}). By the exponential map we define the pre-image of $B^M(q_j,r)$ by $B_r(0)$ and we use the same notation to denote $x\in M$ and its pre-image. We decompose $u_k$ as $u_k=u_{1k}+u_{2k}$, where 	$u_{1k}$ is the solution of
\begin{equation}
\label{2.u1}
\begin{cases}
(-\Delta)^mu_{1k}=\widetilde Qe^{2mu_k}\Xi_r(x), &\mathrm{in}~B_{2r}(0),\\
u_{1k}=\Delta u_{1k}=\cdots=(-\Delta)^{m-1}u_{1k}=0,\quad &\mathrm{on}~\partial B_{2r}(0).
\end{cases}
\end{equation}
where $\Xi_r(x)=dvol_g(x)/dx=1+O(r^2)$ due to the metric tensor $g_{ij}(x)=\delta_{ij}+O(r^2).$ By \cite[Theorem 7]{m2}, we have
\begin{equation}
\label{2.int-est-11}
e^{2m\ell |u_{1k}|}\in L^1(B_{2r}(0))\quad \mbox{for}~\ell\in\left(0,\frac{\Lambda_m}{2\|\widetilde Qe^{2mu_k}\Xi_r(x)\|_{L^1(B_{2r}(0))}}\right)
\end{equation}
and
\begin{equation}
\label{2.int-est}
\int_{B_{2r}(0)}e^{2m\ell|u_{1k}|}dx\leq C(p)r^{2m}.
\end{equation}
Let $G_{r}(x,y)$ be the Green's function of $(-\Delta)^m$ on $B_{2r}(0)$ satisfying the Navier boundary condition, i.e.,
\begin{equation*}
\begin{cases}
(-\Delta)^mG_r(x,y)=\delta_x(y), &\mathrm{in}~B_{2r}(0),\\
G_r(x,y)=\cdots=\Delta^{m-1}G_r(x,y)=0,\quad&\mathrm{on}~\partial B_{2r}(0).
\end{cases}
\end{equation*}
$G_r(x,y)$ can be decomposed as
\begin{equation*}
G_r(x,y)=-\frac{2}{\Lambda_m}\log|x-y|+R_r(x,y)
\end{equation*}
with $R_r(x,y)$ a smooth function for $x,y\in B_{2r}.$ By the Green's representation formula we have
\begin{equation}
\label{2.rep}
u_{1k}(x)=-\frac{2}{\Lambda_m}\int_{B_{2r}(0)}\log|x-y|\widetilde Qe^{2mu_k}\Xi_r(y)dy+O(1),\quad x\in B_{3r/2}(0).
\end{equation}
Observe that
$$
\widetilde Q= d_g(x,q_j)^{2m\alpha_j} \widehat Q
$$
where
\begin{equation}
\label{2.q}
\widehat Q=Q_{g_v}^{2m}e^{-m\Lambda_m\alpha_jR(x,q_j)-m\Lambda_m\sum_{i\neq j}^N\alpha_iG(x,q_i)}~\mbox{is a smooth function in}~B_{2r}(0).
\end{equation}
On the other hand, by using $G(x,y)$ and the Green's representation formula we get that
\begin{equation}
\label{2.repm}
u_k(x)=u_{1k}(x)+u_{2k}(x)=\overline{u}_k+\int_{M}
G(x,y)\widetilde Qe^{2mu_k}\,dvol_g,
\end{equation}
where $\overline{u}_k$ is the average of $u_k$. Since it is known that the leading term of $G$ and $G_{2r}$ carry the same singular potential, we get from Jensen's inequality that
\begin{equation}
\label{2.u2}
u_{2k}=\overline{u}_k+O(1)=\overline{v}_k+O(1)\leq \log\Xint-_{M}e^{v_k}+O(1)=O(1),
\end{equation}
where we used the average of Green function on $M$ is zero. Therefore, we conclude that $u_{2k}$ is bounded uniformly from above. Next, we shall prove that $u_{1k}$ is bounded and our discussion is separated into two cases:

\

\noindent Case 1. $\alpha_j>0$, then \eqref{2.bcon} is equivalent to
\begin{equation*}
\int_{B^M_{2r}(q_j)}\widetilde Qe^{2mu_k}\,dvol_g<\frac12\Lambda_m.
\end{equation*}
Since $u_{2k}$ is bounded from above in $B_{3r/2}(0)$, we see from  \eqref{2.int-est} that there exists some $\ell>1$ such that $\widetilde Qe^{2mu_k}\in L^\ell(B_{2r}^M(q_1))$. It is easy to see that $u_k\in L^1(M)$. Together with \eqref{2.rep} we can easily see that $u_{1k},u_{2k}\in L^1(B_{3r/2}(0)).$ By the interior regularity results in \cite[Theorem 1]{b} we get that
\begin{equation}
\label{2.est-b}
\|u_{1k}\|_{W^{2m,\ell}(B_{r}(0))}\leq \|\widetilde Qe^{2mu_k}\|_{L^\ell(B_{3r/2}(0))}+\|u_{1k}\|_{L^{1}(B_{r}(0))}\leq C.
\end{equation}
Thus by the classical Sobolev inequality we get that $u_{1k}\in L^\infty(B_r(0))$.

\medskip

\noindent Case 2. If $\alpha_j\in(-1,0)$ then \eqref{2.bcon} is equivalent to
\begin{equation*}
\int_{B_{2r}^M(q_j)}\widetilde Qe^{2mu_k}\,dvol_g<\frac12\Lambda_m(1+\alpha_j).
\end{equation*}
It is not difficult to see $|x|^{2m\alpha_j}\in L^{\ell}(B_{2r})$ for any $\ell\in[1,-\frac{1}{\alpha_j})$ and $e^{2mu_{1k}}\in L^p(B_{2r})$ for $p\in[1,\frac{1}{1+\alpha_1}+\epsilon)$ for some small strictly positive number $\epsilon$ by \eqref{2.int-est-11}. As a consequence, we get that $|x|^{2m\alpha_1}e^{2mu_{1k}}\in L^{\ell}(B_{2r}(0))$ for some $\ell>1$ by H\"older inequality. Repeating the arguments as in Case 1,  we obtain that $u_{1k}$ is bounded uniformly in $B_r(0)$.

After establishing that $u_{1k}$ is bounded in $B_r(0)$, combining with \eqref{2.u2} we derive that $u_k$ is bounded above in $B_r^M(q_j)$. Then we finish the proof of this lemma.
\end{proof}

\medskip

We shall derive now the quantization result and the concentration property of the bubbling solution.

\begin{proposition}
Let $\{u_k\}$ be a sequence of solution to \eqref{2.u} and $\mathcal{B}$ be its blow-up set, then we have the following convergence in the sense of measures
\begin{equation}
\widetilde Q e^{2mu_k}\rightharpoonup \sum_{p\in\mathcal{B}}\Lambda_m(1+\alpha_p)\delta_p,\quad \mbox{as}~k\to+\infty,
\end{equation}
where
\begin{equation*}
\alpha_p=\begin{cases}
\alpha_i,\quad &\mbox{if}\quad p=q_i\in\{q_1,\cdots,q_N\},\\
0,\quad  &\mbox{if}\quad  p\in \mathcal{B}\setminus\{q_1,\cdots,q_N\}	.
\end{cases}
\end{equation*}
In particular, $u_k\to-\infty$ uniformly on any compact subset of $M\setminus \mathcal{B}.$
\end{proposition}

\begin{proof}
For any compact set $K\subset M\setminus\mathcal{B}$, we can use the Green's representation formula
\begin{equation}
\label{2.p-1}
u_k(x)-u_k(y)=\int_{M}\left(G(x,z)-G(y,z)\right)\left(\widetilde Qe^{2mu_k}-Q_{g}^{2m}\right)dz
\end{equation}
together with the estimate \eqref{2.est-g} to derive that
\begin{equation}
\label{2.p-2}
|\nabla^i u_k(x)|\leq C(K)~\mbox{for}~x\in K,\quad  1\leq i\leq 2m-1.
\end{equation}
Then from equation \eqref{2.u} and classical elliptic estimates  we get the $2m$ order derivatives of $u_k$
\begin{equation}
\label{2.p-3}
|\nabla^{2m}u_k(x)|\leq C(K)~\mbox{for}~x\in K.
\end{equation}

To proceed with our discussion we introduce the following quantity
\begin{equation}
\label{2.p-l}
\sigma_p=\lim_{r\to0}\lim_{k\to+\infty}\int_{B^M(p,r)}\widetilde Qe^{2mu_k}\,dvol_g.
\end{equation}
It has been shown in Lemma \ref{le2.2} that $\sigma_p$ has a positive lower bound at the blow-up point. From the fact that $\int_M Q_g^{2m}\,dvol_g$ is finite, we conclude that the blow-up points are finite. At a regular blow-up point $p$ it has been already shown in \cite[Theorem 2]{m1} that
\begin{equation*}
\widetilde Qe^{2mu_k}\rightharpoonup \Lambda_m\delta_p \quad\mathrm{in}\quad B^M_{r_p}(p),
\end{equation*}
where $r_p$ is chosen such that $B^M(p,r_p)\cap(\mathcal{B}\setminus\{p\})=\emptyset$. In the following discussion  we will focus on the singular blow-up point. Without loss of generality, we shall consider $u_k$ in $B^M_{2r}(q_1)$, where $r$ is chosen such that $B^M_{2r}(q_1)$ only contains $q_1$ from $\mathcal{B}$. We first claim that
\begin{equation}
\label{2.p.claim}
u_k\to-\infty\quad\mbox{for}\quad x\in B_{2r}^M(q_1)\setminus\{q_1\}.
\end{equation}
We prove it by contradiction. Suppose that $u_k$ is uniformly bounded below at some point away from $q_1$. Then by \eqref{2.p-2} we derive that
\begin{equation}
\label{2.p.conv}
u_k\to u_0\quad\mbox{in}\quad C_{\mathrm{loc}}^{2m-1,\sigma}(B^M_{2r}(q_1)\setminus\{q_1\}),	\quad \sigma\in(0,1),
\end{equation}
with the limit function verifying
\begin{equation}
P_g^{2m}u_0+Q_g^{2m}=d_g(x,q_1)^{2m\alpha_1}\widehat Qe^{2mu_0}\quad \mathrm{in}\quad B^M_{2r}(q_1)\setminus \{q_1\}.
\end{equation}
where $\widehat Q$ is a smooth function around $q_1$ defined analogously as in \eqref{2.q} and $d_g(x,q_1)$ denotes the geodesic distance between $x$ and $q_1$ with respect to the metric $g$. According to the definition $\sigma_p$ (see \eqref{2.p-l}) we see that $u_0$ satisfies
\begin{equation*}
P_g^{2m}u_0+Q_g^{2m}=d_g(x,q_1)^{2m\alpha_1}\widehat Qe^{2mu_0}+\sigma_{q_1}\delta_{q_1}\quad \mbox{in}\quad B_{2r}^M(q_1).
\end{equation*}
Using the Green's representation formula for $u_0$, we have
\begin{equation}
\label{2.p-re}
u_0(x)=\sigma_{q_1}G(x,q_1)+v_0(x),
\end{equation}
where
\begin{equation}
\label{2.p-v}
\begin{aligned}
v_0(x)=&\int_M\frac{2}{\Lambda_m}\log d_g(x,y)\left(d_g(y,q_1)^{2m\alpha_1}\widehat Qe^{2mu_0}\right)\,dvol_g\\
&+\int_MR(x,y)\left(d_g(y,q_1)^{2m\alpha_1}\widehat Qe^{2mu_0}\right)\,dvol_g.
\end{aligned}
\end{equation}
Denoting the two terms on right hand side by $\hat v_1$ and $\hat v_2$ respectively, it is not difficult to see that $\hat v_2$ is smooth. In the following we shall prove that $\hat v_1(x)$ is bounded in $x\in B^M_r(q_1)$. In fact, for $x\in B^M_r(q_1)$ we have
\begin{equation}
\label{2.p-low}
\begin{aligned}
v_0(x)~&=\int_{B^M_r(q_1)}G(x,y)d_g(y,q_1)^{2m\alpha_1}\widehat Qe^{2mu_0}\,dvol_g+O(1),\\
&\geq\frac{2}{\Lambda_m}\log\frac{1}{r}\|\widehat Q|x|^{2m\alpha_1}e^{2mu_0}\|_{L^1(B^M_{2r}(q_1))}+O(1).
\end{aligned}
\end{equation}
This provides a lower bound for $v_0(x)$. On the other hand, we have
\begin{equation}
\label{2.p.lb}
d_g(x,q_1)^{2m\alpha_1}e^{2mu_0}
\geq Cd_g(x,q_1)^{2m(\alpha_1-\frac{2\sigma_{q_1}}{\Lambda_m})}.
\end{equation}
Using the fact that the left hand side of \eqref{2.p.lb} is integrable we get
\begin{equation}
\label{2.ine-alpha}
\alpha_1-\frac{2\sigma_{q_1}}{\Lambda_m}>-1.
\end{equation}
When $\alpha_1<0$, we see that the above inequality \eqref{2.ine-alpha} implies that $\sigma_{q_1}<\Lambda_m(1+\alpha_1)$. Then $u_k$ can not blow-up at $q_1$ by Lemma \ref{le2.2} and we get a contradiction. This implies \eqref{2.p.claim} for $\alpha_1<0$. For $\alpha_1>0$ we have
\begin{equation}
\label{2.p.lb-2}
Cd_g(x,q_1)^{2m\alpha_1-\frac{4m\sigma_{q_1}}{\Lambda_m}}
e^{v_0(x)}
\geq d_g(x,q_1)^{2m\alpha_1}e^{2mu_0}\geq
Cd_g(x,q_1)^{2m\alpha_1-\frac{4m\sigma_{q_1}}{\Lambda_m}}.
\end{equation}
In order to show that $\hat v_1(x)$ is bounded in $B_r^M(q_1)$, we study $\hat v_1(x)$ in terms of local coordinates at $q_1$. Then $d_g(x,q_1)$ can be regarded as $|x^p-0|$, where $x^p$ denotes its pre-image of $x$ under the exponential map at $q_1$. By a little abuse of notation, we still denote $x^p$ by $x$. Then we notice that $\hat v_1(x)$ satisfies
\begin{equation}
\label{2.p.eq}
(-\Delta)^{m}\hat v_1(x)=|x|^{2m\alpha_1}\widehat Qe^{2mu_0}\Xi(x)\quad\mbox{in}\quad B_{2r}(0),
\end{equation}
where $\Xi(x)=\frac{dvol_g(x)}{dx}$ is bounded above and below in $B_{2r}(0)$ since the metric tensor is comparable to the standard Euclidean metric. Since we only consider the local behavior of $\hat v_1(x)$ in $B_r(0)$, by multiplying a cut-off function $\chi(x)$ with $\chi(x)=1$ for $|x|\leq r$ and $\chi(x)=0$ for $|x|\geq 2r$, we have $\tilde v_1(x):=\chi(x)\hat v_1(x)$ verifies
\begin{equation}
\label{2.eq-v1}
\begin{cases}
(-\Delta)^m\tilde v_1(x)=|x|^{2m\alpha_1}\widehat Qe^{2mu_0}\Xi(x)+\Xi_0(x)
\quad  &\mbox{in}\quad  B_{2r}(0),\\
\\
\tilde v_1(x)=\Delta \tilde v_1(x)=\cdots=\Delta^{m-1}\tilde v_1(x)=0 \quad  &\mbox{on}\quad \partial B_{2r}(0),
\end{cases}
\end{equation}
where $\Xi_0(x)$ is smooth in $B_{2r}(0)$. It is not difficult to see that
$$|x|^{2m\alpha_1}\widehat Qe^{2mu_0}\Xi(x)+\Xi_0(x)\in L^1(B_2r(0)).$$
We decompose $|x|^{2m\alpha_1}\widehat Qe^{2mu_0}\Xi(x)+\Xi_0(x)$ into $P_1(x)$ and $P_2(x)$ with
\begin{equation}
\label{2.p-condition}
\|P_1\|_{L^1(B_{2r}(0))}\leq \varepsilon\quad \mathrm{and}\quad P_2\in L^\infty(B_{2r}(0)).
\end{equation}
Correspondingly we decompose $\tilde v_1$ into $\tilde v_{11}$ and $\tilde v_{12}$, where $\tilde v_{1i},~i=1,2$ solve
\begin{equation}
\label{2.eq-v1i}
\begin{cases}
(-\Delta)^m\tilde v_{1i}(x)=P_i(x)
\quad  &\mbox{in}\quad B_{2r}(0),\\
\\
\tilde v_{1i}(x)=\Delta \tilde v_{1i}(x)=\cdots=\Delta^{m-1}\tilde v_{1i}(x)=0 \quad  &\mbox{on}\quad \partial B_{2r}(0).
\end{cases}
\end{equation}
For $\tilde v_{11},$ by \cite[Theorem 7]{m2} we get that
$e^{\frac{\Lambda_m}{2\varepsilon}\tilde v_{11}}\in L^1(B_{2r}(0)).$
While for $\tilde v_{12}$, using the classical elliptic regularity theory we derive that $\tilde v_{12}\in L^\infty(B_{2r}(0))$. Together with  \eqref{2.p.lb-2} we can select $\varepsilon$ sufficiently small such that
$$|x|^{2m\alpha_1}\widehat Qe^{2mu_0}\Xi(x)\in L^l(B_{2r}(0))\quad \mbox{for some}\quad l>1.$$
Returning to equation \eqref{2.eq-v1} we apply the regularity theory to deduce that $\hat v_1\in W^{2m,l}(B_{2r}(0))$. As a consequence, we have $v_0\in W^{2m,l}(B_{2r}(0))$ and it implies that $|v_0|\leq  C$ for some constant $C$ in $B_{r}(0)$ by the classical Sobolev inequality. Thus we have proved $v_0$ is bounded in $B^M_r(q_1)$. Together with \eqref{2.p.lb-2} we get that
\begin{equation}
\label{2.p-bb}
d_g(x,q_1)^{2m\alpha_1}\widehat Qe^{2mu_0}
\sim{d_g(x,q_1)^{2m(\alpha_1-\frac{2\sigma_{q_1}}{\Lambda_m})}}\quad \mbox{if}\quad  \alpha_1>0.
\end{equation}
Next we shall derive a contradiction by making use of the Pohzozaev identity. It is known that equation \eqref{2.u} can be written as
\begin{align}
\label{2.i-0}
(-\Delta_g)^{m}u_k(x)+\mathcal{A}u_k+Q_g^{2m}=\widetilde Qe^{2mu_k}\quad \mathrm{in}\quad B_r^M(q_1),
\end{align}
where $\mathcal A$ is a linear differential operator of order at most $2m-1$, moreover the coefficients of $A$ belong to $C_{\mathrm{loc}}^l(M)$ for all $l\geq 0$. Using the local normal coordinate, by Lemma \ref{lea.m} (see also Remark \ref{rem-local}) we could write \eqref{2.i-0} as
\begin{equation}
\label{2.i-1}
(-\Delta)^mu_k+\mathcal{D}^{2m}u_k+\mathcal{C}u_k+Q_g^{2m}=\widetilde Qe^{2mu_k}\quad \mathrm{in}\quad B_r(0),
\end{equation}
where $\mathcal{D}^{2m}$ is a linear differential operator of order $2m$ and the coefficients are of order $O(|x|^2)$ with its derivative of arbitrary order smooth, while $\mathcal{C}$ is a linear differential operator of order at most $2m-1$, and the coefficients of $B_k$ belong to $C_{\mathrm{loc}}^l(\mathbb{R}^{2m})$ for all $l\geq 0$. Multiplying $x\cdot\nabla u_k$ on both sides, concerning the right hand side, we have
\begin{equation}
\label{2.i-r}
\begin{aligned}
\mbox{R.H.S. of }\eqref{2.i-1}=~&\frac{1}{2m}\int_{B_r(0)}x\cdot\nabla (|x|^{2m\alpha_1}\widehat Qe^{2m u_k})\,dx
-\alpha_1\int_{B_r(0)}|x|^{2\alpha_1}\widehat Qe^{2m u_k}\,dx\\
&-\frac{1}{2m}\int_{B_r(0)}(x\cdot\nabla\widehat Q)|x|^{2m\alpha_1}e^{2m u_k}\,dx\\
=~&\int_{\partial B_r(0)}|x|^{2\alpha_1+1}\widehat Qe^{2m u_k}\,ds
-(\alpha_1+1)\int_{B_r(0)}|x|^{2\alpha_1}\widehat Qe^{2m u_k}\,dx\\
&-\frac{1}{2m}\int_{B_r(0)}(x\cdot\nabla\widehat Q)|x|^{2m\alpha_1}e^{2m u_k}\,dx\\
\to~&-(1+\alpha_1)\sigma_{q_1}+o_r(1)\quad\mathrm{as}\quad  k\to+\infty.
\end{aligned}
\end{equation}
Next, we consider the left hand side of \eqref{2.i-1}. At first, for the fourth term, we have
\begin{equation}
\label{2.i-l-3}
\left|\int_{B_r(0)}\langle x,\nabla u_k\rangle Q_g^{2m}dx\right|
\leq r\int_{B_r(0)}|\nabla u_k|dx\leq Cr^{2m},
\end{equation}
where we used Lemma \ref{le2.1}. Therefore
\begin{equation}
\label{2.i-l-3-1}
\lim_{r\to0}\lim_{k\to+\infty}\left|\int_{B_r(0)}\langle x,\nabla u_k\rangle Q_g^{2m}dx\right|=0.
\end{equation}
For the third term, we have
\begin{equation}
\label{2.i-l1}
\begin{aligned}
\left|\int_{B_r(0)}\langle x,\nabla u_k\rangle\mathcal{C}u_k\right|
\leq ~&C\sum_{i=0}^{2m-1}\int_{B_r(0)}|x||\nabla^iu_k||\nabla u_k|dx\\
\leq~&\sum_{i=1}^{2m-1}
\left(\int_{B_r(0)}|x|^{s_i}|\nabla u_k|^{s_i}dx\right)^{\frac{1}{s_i}}\left(\int_{B_r(0)}|\nabla^iu_k|^{t_i}dx\right)^{\frac{1}{t_i}}\\
&+C\int_{B_r(0)}|\nabla u_k|dx,
\end{aligned}
\end{equation}
where we used $|x| |u_k|\leq C$ in $B_r(0)$,
$$t_i=\frac{2m}{i}-\delta\quad\mbox{and}\quad
s_i=\frac{2m-\delta i}{2m-i-\delta i},\qquad \delta\in\left(0,\frac{1}{2(2m-1)}\right).$$
From \eqref{2.p-re}, \eqref{2.p-v} and since $\hat v_1,~\hat v_2$ are bounded from above, we can see that $|x\cdot\nabla u_0|\leq C$ in $B_r(0)$. Together with Lemma \ref{le2.1} we have
\begin{equation}
\label{2.i-l2}
\int_{B_r(0)}|x|^{s_i}|\nabla u_0|^{s_i}dx\leq Cr^{2m},
\quad\mbox{and}\quad
\begin{aligned}
\int_{B_r(0)}|\nabla^iu_0|^{t_i}dx\leq Cr^{i\delta}.
\end{aligned}
\end{equation}
As a consequence of \eqref{2.i-l1} and \eqref{2.i-l2}, we see that
\begin{equation}
\label{2.i-l-2}
\lim_{r\to0}\lim_{k\to+\infty}\left|\int_{B_r(0)}\langle x,\nabla u_k\rangle\mathcal{C}u_k\right|\,dx=0.
\end{equation}
For the second term, we have already seen that $v_0\in W^{2m,l}(B_r(0))$ for some $l>1$, then using \eqref{2.p-re}, we see that
\begin{equation*}
\int_{B_r(0)}x\cdot\nabla u_0 \mathcal{D}^{2m}u_0\,dx
\leq C\int_{B_r(0)}\frac{1}{|x|^{m-2}}\,dx+Cr^2\|v_0\|_{W^{2m,1}(B_{r}(0))}\leq Cr^2.
\end{equation*}
It leads to
\begin{equation}
\label{2.i-l-22}
\lim_{r\to0}\lim_{k\to+\infty}\left|\int_{B_r(0)}\langle x,\nabla u_k\rangle\mathcal{D}^{2m}u_k\,dx\right|=0.
\end{equation}
Therefore, from \eqref{2.i-l-3-1}, \eqref{2.i-l-2} and \eqref{2.i-l-22} we get that except from the first on the left hand side of \eqref{2.i-1}, the other terms vanish in the limit. It remains to study the term $\int_{B_r(0)}(-\Delta)^mu_k x\cdot\nabla u_kdx$. We shall only consider the case when $m$ is even, the argument for the case $m$ is odd goes almost the same. We set $m=2m_0$. Using the Pohozaev identity\eqref{a.poho1}, replacing $f$ by $\frac{2\sigma_{q_1}}{\Lambda_m}\log|x|$ plus a smooth function, after direct computations we get that
\begin{equation}
\label{2.il-1}
\begin{aligned}
&\int_{B_r(0)}(-\Delta)^mu_0\langle x,\nabla u_0\rangle dx\\
&=\sum_{i=2}^{m_0}\int_{\partial B_r(0)}2^{2m}(m-1)!(m-1)!\left(1-\frac{i-1}{m-i}\right)\frac{\sigma_{q_1}^2}{\Lambda_m^2}\frac{1}{r^{2m-1}}ds\\
&\quad +\sum_{i=1}^{m_0}\int_{\partial B_r(0)}2^{2m}(m-1)!(m-1)!\left(\frac{i-1}{m-i}-1\right)\frac{\sigma_{q_1}^2}{\Lambda_m^2}\frac{1}{r^{2m-1}}ds\\
&\quad -\int_{\partial B_r(0)}2^{2m-1}(m-1)!(m-1)!\frac{\sigma_{q_1}^2}{\Lambda_m^2}\frac{1}{r^{2m-1}}ds+o_r(1)
\\
&\to -2^{2m-1}(m-1)!(m-1)!\frac{\sigma_{q_1}^2}{\Lambda_m^2}|\mathbb{S}^{2m-1}|\quad \mathrm{as}\quad r\to0.
\end{aligned}
\end{equation}
It is known that we can write
\begin{equation*}
\Lambda_m=(2m-1)!|\mathbb{S}^{2m}|=2^{2m-1}(m-1)!(m-1)!|\mathbb{S}^{2m-1}|.
\end{equation*}
Together with \eqref{2.i-r} and \eqref{2.il-1} we derive that
\begin{equation}
\label{2.i-final}
(1+\alpha_1)\sigma_{q_1}=\frac{\sigma_{q_1}^2}{\Lambda_m}.
\end{equation}
Recalling also Lemma \ref{le2.2} we get
\begin{equation}
\label{2.i-final-1}
\sigma_{q_1}=(1+\alpha_1)\Lambda_m.
\end{equation}
Returning to equation \eqref{2.p-bb} we see that
\begin{equation}
d_g(x,q_1)^{2m\alpha_1}\widehat Qe^{2mu_0}
\sim d_g(x,q_1)^{-2m-2m(1+\alpha_1)}\geq d_g(x,q_1)^{-2m},
\end{equation}
which contradicts  	$d_g(x,q_1)^{2m\alpha_1}\widehat Qe^{2mu_0}\in L^1(B_r(0))$. Therefore, $u_k\to-\infty$ uniformly on any compact subset of $B^M_{2r}(q_1)\setminus\{q_1\}$.

\medskip

It remains to show the quantization $\sigma_{q_1}$ is exactly $\Lambda_m(1+\alpha_1)$. We consider the function $\hat u_k(x)=u_k(x)-c_k$, with $c_k=\Xint-_{\partial B_r(0)}u_k(x)\to-\infty$. As before, we can show that $\hat u_k(x)$ converges to some function $\hat u_0(x)$ in $C_{\mathrm{loc}}^{2m}(B_{2r}\setminus\{0\})$ and we can write
$$\hat u_0(x)=-\frac{2\sigma_{q_1}}{\Lambda_m}\log d_g(x,q_1)+\hat v(x).$$
Repeating the previous argument, again by the Pohozaev identity we derive that $\sigma_{q_1}=\Lambda(1+\alpha_1)$ and we finish the proof.
\end{proof}

\

\section{Existence result}

In this section we are going to prove the existence and multiplicity results of Theorems \ref{th1.2} and \ref{th-mult}. To make the exposition more transparent we assume hereafter for simplicity $P_g^{2m}\geq0$. The general case can be treated by suitable adaptations, see Remark \ref{rem-general}.

\medskip

Solutions of \eqref{1.mliouville} are critical points of the functional
\begin{equation}\label{functional}
\mathcal E(u)=2m \int_M uP^{2m}_gu\, dvol_g +4m \int_M \Bigr(Q^{2m}_g+\dfrac{\Lambda_m}{2|M|} \sum_{j=1}^N \alpha_j  \Bigr)u\,dvol_g-2\kappa_{g_v} \log \int_{M} \widetilde Qe^{2mu}\,dvol_g
\end{equation}
with $u \in H^{m}(M)$, where we recall
$$
	\widetilde Q=  Q_{g_{v}}^{2m}e^{-m\Lambda_m\sum_{j=1}^N\alpha_jG(x,q_j)}>0, \quad \Lambda_m=(2m-1)!|\mathbb{S}^{2m}|
$$
and
$$
  \kappa_{g_v}=\int_M\widetilde Qe^{2mu}\,dvol_g=\int_MQ_{g}^{2m}\,dvol_{g}+\dfrac{\Lambda_m}{2} \sum_{j=1}^N \alpha_j.
$$
We point out we consider here
$$
\alpha_j>0 \quad \forall j=1,\dots,N.
$$
Then, in particular, the Adams-Trudinger-Moser inequality in Theorem \ref{ATM} implies the functional $\mathcal E$ is coercive and bounded from below provided $\kappa_{g_v}<\Lambda_m$. Thus, existence of solutions in this subcritical case follows by direct method of calculus of variations.

\medskip

In the supercritical case $\kappa_{g_v}>\Lambda_m$ the functional is unbounded from below and we need to apply a min-max method based on the  topology of the sublevels of the functional
$$
	\mathcal E^a =\bigr\{ u\in H^m(M) \,:\, \mathcal E(u)\leq a \bigr\}.
$$
The rough idea is that the low sublevels carry some non-trivial topology while the high sublevels are contractible and such change of topology jointly with the compactness property of Theorem~\ref{th1.1} (provided $\kappa_{g_v}\notin\Gamma$) detects a critical point. The main step is the study of low sublevels which is done by improved version of the Adams-Trudinger-Moser inequality and suitable test functions.

\medskip

Let us start by pointing out that once the Adams-Trudinger-Moser inequality is available (Theorem~\ref{ATM}), then a standard argument yields improved versions of it under a spreading of the conformal volume $\widetilde Qe^{2mu}$ in, say, $l$ disjoint subsets as expressed in \eqref{spread}. Somehow, it is possible to sum up localized versions of the inequality which are in turn based on cut-off functions and improve the Adams-Trudinger-Moser constant to roughly $l\Lambda_m$. We refer the interested readers for example to Lemma 4.1 in \cite{ndiaye} and references therein.
\begin{lemma} \label{lem-improved}
Let $\delta,\theta>0$, $l\in\mathbb{N}$ and $\Omega_1,\dots,\Omega_l\subset M$ be such that $d(\Omega_i,\Omega_j)>\delta$ for any $i\neq j$. Then, for any $\varepsilon>0$ there exists $C=C(\varepsilon,\delta,\theta,L,M)$ such that for any $u\in H^m(M)$ such that
\begin{equation} \label{spread}
	\int_{\Omega_i}\dfrac{\widetilde Qe^{2mu}\,dvol_g}{\int_M \widetilde Q\,e^{2mu}\,dvol_g} \geq \theta \qquad \forall i\in\{1,\dots, l\}
\end{equation}
it holds
$$
l\Lambda_m\log \int_M \widetilde Qe^{2m(u-\bar u)}\,dvol_{g} \leq (1+\varepsilon)m \int_M uP^{2m}_g u\,dvol_{g} + C,
$$
where $\bar u$ is the average of $u$.
\end{lemma}

Improved inequalities then yield improved lower bounds on the functional $\mathcal E$. As a consequence, in the low sublevels $\widetilde Qe^{2mu}$ has to be concentrated in not too many different subsets, as shown in the following result.
\begin{lemma}\label{lem-bound}
Suppose $\kappa_{g_v}<(k+1)\Lambda_m$ for some $k\in\mathbb{N}$. Then, $\forall\varepsilon,r>0$ there exists $L=L(\varepsilon,r)\gg1$ such that $\forall u\in\mathcal E^{-L}$ there exist $k$ points $\{p_1,\dots,p_k\}\subset M$ such that
$$
	\int_{\cup_{i=1}^k B_r^M(p_i)}\dfrac{\widetilde Qe^{2mu}\,dvol_g}{\int_M \widetilde Q\,e^{2mu}\,dvol_g} \geq 1-\varepsilon.
$$
\end{lemma}
\begin{proof}
We sketch here the proof. Suppose the thesis is false. Then, using a standard covering argument as in Lemma 2.3 of \cite{dm} there exist $k+1$ disjoint subsets $\Omega_1,\dots,\Omega_{k+1}\subset M$ in which $\widetilde Qe^{2mu}$ is spread in the sense of \eqref{spread}. Therefore, applying the improved Adams-Trudinger-Moser inequality of Lemma \ref{lem-improved} we would get a lower bound of the functional
$$
	\mathcal E(u)\geq 2m\left( 1-\dfrac{\kappa_{g_v}}{(k+1)\Lambda_m}(1+\varepsilon) \right)\int_M uP^{2m}_g u\,dvol_{g} + {l.o.t.}
$$
By assumption $\kappa_{g_v}<(k+1)\Lambda_m$ and hence we can take a sufficiently small $\varepsilon>0$ such that
$$
1-\frac{\kappa_{g_v}}{(k+1)\Lambda_m}(1+\varepsilon)\geq0
$$
which yields $\mathcal E(u)\geq-L$ for some $L\gg1$. This is not possible since we were considering $u\in\mathcal E^{-L}$.
\end{proof}

\medskip

It is then convenient to describe the low sublevels by means of formal barycenters of $M$ of order $k$, that is unit measures supported in at most $k$ points on $M$, defined by
\begin{equation} \label{bary}
	M_k = \left\{ \sum_{i=1}^k t_i\delta_{p_i} \,:\, \sum_{i=1}^k t_i=1,\,t_i\geq0,\,p_i\in M, \ \forall i=1,\dots,k \right\}.
\end{equation}
The idea is to use a projection within unit measures such that
$$
\dfrac{\widetilde Qe^{2mu}\,dvol_g}{\int_M \widetilde Q\,e^{2mu}\,dvol_g} \mapsto \sigma\in M_k .
$$
This is done exactly as in Proposition 3.1 of \cite{dm} by using Lemma \ref{lem-bound} to get the following.
\begin{proposition}
Suppose $\kappa_{g_v}<(k+1)\Lambda_m$ for some $k\in\mathbb{N}$. Then, there exists $L\gg1$ and a projection $\Psi:\mathcal E^{-L}\to M_k$.
\end{proposition}
Recall now that we are assuming there exists a retraction $R:M\to M^{\mbox{\tiny{R}}}$ with $M^{\mbox{\tiny{R}}}\subset M$ a closed $n$-dimensional submanifold, $n\in[1,2m]$, such that $\a_j\notinM^{\mbox{\tiny{R}}}$ for all $j=1,\dots,N$. Let $M^{\mbox{\tiny{R}}}_k$ be the set of formal barycenters of $M^{\mbox{\tiny{R}}}$. We can then define a map $\Psi_R:\mathcal E^{-L}\to M^{\mbox{\tiny{R}}}_k$ simply by considering the composition
$$
	\mathcal E^{-L} \stackrel{\Psi}{\longrightarrow} M_k \stackrel{R_*}{\longrightarrow}  M^{\mbox{\tiny{R}}}_k,
$$
where $R_*$ is the push-forward of measures induced by the retraction $R$. Therefore, we have the following result.
\begin{lemma} \label{lem-psi}
Suppose $\kappa_{g_v}<(k+1)\Lambda_m$ for some $k\in\mathbb{N}$. Then, there exists $L\gg1$ and a continuous map $\Psi_R:\mathcal E^{-L}\to M^{\mbox{\tiny{R}}}_k$.
\end{lemma}

\medskip

The low sublevels are thus naturally described (at least partially) by $M^{\mbox{\tiny{R}}}_k$. As a matter of fact, we are going to construct a reverse map, mapping continuously $M^{\mbox{\tiny{R}}}_k$ into $\mathcal E^{-L}$. This is done by suitable test functions on which the functional attains low values. The idea here is that, since $M^{\mbox{\tiny{R}}}$ does not contain conical points $q_j$, we may consider a family of \emph{regular} bubbles centered on $M^{\mbox{\tiny{R}}}$. We thus take a non-decreasing cut-off function $\chi_\delta$ such that
$$
\begin{cases}
	\chi_\delta(t)=t, \quad &t\in[0,\delta], \\
	\chi_\delta(t)=2\delta, \quad &t\geq2\delta,
\end{cases}
$$
let $\lambda>0$ and then define $\Phi:M^{\mbox{\tiny{R}}}_k\to H^m(M)$ by
$$
	\Phi(\sigma)=\varphi_{\lambda,\sigma}, \quad \sigma=\sum_{i=1}^k t_i\delta_{p_i}\inM^{\mbox{\tiny{R}}}_k,
$$
where
$$
	\varphi_{\lambda,\sigma}(y)=\frac{1}{2m}\log\sum_{i=1}^k t_i \left( \dfrac{2\lambda}{1+\lambda^2\chi_\delta^2(d(y,p_i))} \right)^{2m}.
$$
Now, since we are considering bubbles centered on $M^{\mbox{\tiny{R}}}$ which does not contain conical points, we can neglect the effect of the singularities and all the following estimates are carried out exactly as in the regular case, that is Lemma 4.5, Lemma~4.6 in \cite{ndiaye} and  references therein. To avoid technicalities, with a little abuse of notation we will write $o(1)$ to denote quantities which do not necessarily tend to zero but that can be made arbitrarily small.
\begin{lemma} \label{lem-est}
Let $\varphi_{\lambda,\sigma}$ be as above. Then, for $\lambda\to+\infty$ it holds
\begin{align*}
	\int_M \varphi_{\lambda,\sigma}P^{2m}_g \varphi_{\lambda,\sigma}\,dvol_{g} & \leq  2k\Lambda_m(1+o(1)) \log\lambda,\\
	\int_M \Bigr(Q^{2m}_g+\dfrac{\Lambda_m}{2|M|} \sum_{j=1}^N \alpha_j  \Bigr) \varphi_{\lambda,\sigma}\,dvol_g & = -\kappa_{g_v}(1+o(1)) \log\lambda,\\
	\log\int_M \widetilde Qe^{2m\varphi_{\lambda,\sigma}}\,dvol_g & = O(1).
\end{align*}
\end{lemma}

\medskip

By the latter estimates we readily get the map we were looking for if we take $\kappa_{g_v}>k\Lambda_m$. Indeed, it is enough to observe that by Lemma \ref{lem-est} we have
$$
	\mathcal E(\Phi(\sigma)) \leq 4m(k\Lambda_m-\kappa_{g_v}+o(1))\log\lambda \to -\infty
$$
as $\lambda\to+\infty$. Therefore, we can state the following result.
\begin{proposition} \label{prop-phi}
Suppose $\kappa_{g_v}>k\Lambda_m$ for some $k\in\mathbb{N}$. Then, for any $L>0$ there exists $\lambda\gg1$ such that $\Phi:M^{\mbox{\tiny{R}}}_k\to\mathcal E^{-L}$.
\end{proposition}

\

We are now in position to prove the existence result.
\begin{proof}[Proof of Theorem \ref{th1.2}]
Suppose $\kappa_{g_v}\in(k\Lambda_m,(k+1)\Lambda_m)$ for some $k\in\mathbb{N}$ and $\kappa_{g_v}\notin\Gamma$, where $\Gamma$ is the critical set given in \eqref{gamma}. The proof is based on a min-max argument relying on the set $M^{\mbox{\tiny{R}}}_k$ which will keep track of the topological properties of the low sublveles of the functional $\mathcal E$, jointly with the compactness property in Theorem~\ref{th1.1}.

\

\noindent \emph{Step 1.} Recalling Lemma \ref{lem-psi}, let $L\gg1$ be such that there exists a continuous map $\Psi_R:\mathcal E^{-L}\to M^{\mbox{\tiny{R}}}_k$. Then, by Proposition \ref{prop-phi} we can take $\lambda\gg1$ such that $\Phi:M^{\mbox{\tiny{R}}}_k\to\mathcal E^{-L}$. Consider now the composition
$$\begin{array}{ccccl}
M^{\mbox{\tiny{R}}}_k&\stackrel{\Phi}{\longrightarrow}&\mathcal E^{-L}&\stackrel{\Psi_{R}}{\longrightarrow}& M^{\mbox{\tiny{R}}}_k \hspace{2cm} \\
\sigma&\mapsto&\varphi_{\lambda,\sigma}&\mapsto&\Psi_{R}\left(\frac{\widetilde Qe^{2m\varphi_{\lambda,\sigma}}\,dvol_g}{\int_M \widetilde Q\,e^{2m\varphi_{\lambda,\sigma}}\,dvol_g}\right).
\end{array}$$
It is not difficult to see that the latter composition is homotopic to the identity map on $M^{\mbox{\tiny{R}}}_k$. We have just to notice that, as $\lambda\to+\infty$, $\frac{\widetilde Qe^{2m\varphi_{\lambda,\sigma}}\,dvol_g}{\int_M \widetilde Q\,e^{2m\varphi_{\lambda,\sigma}}\,dvol_g}\rightharpoonup\sigma$ in the sense of measures, that $\Psi$ is a projection and that $R$ is a retraction onto $M^{\mbox{\tiny{R}}}$. The homotopy is thus realized by letting  $\lambda\to+\infty$. As a consequence, if we consider the induced maps between homology groups $H_*$ we get that
\begin{equation} \label{inj}
H_*(M^{\mbox{\tiny{R}}}_k) \hookrightarrow H_*\bigr(\mathcal E^{-L}\bigr) \mbox{ injectively}.
\end{equation}
Now, since $M^{\mbox{\tiny{R}}}$ is a closed manifold, it is well-known that $M^{\mbox{\tiny{R}}}_k$ has non-trivial homology groups and hence, in particular, it is non-contractible. We refer the interested readers for example to Lemma 3.7 in \cite{dm} where the $4$-dimensional case is considered. By the above discussion, this implies
\begin{equation}\label{non-contr}
	\Phi(M^{\mbox{\tiny{R}}}_k)\subset \mathcal E^{-L} \mbox{ is non-contractible}.
\end{equation}

\medskip

\noindent \emph{Step 2.} We next consider the topological cone over $M^{\mbox{\tiny{R}}}_k$ which is defined through the equivalence relation
$$
	\mathcal C = \dfrac{M^{\mbox{\tiny{R}}}_k\times[0,1]}{M^{\mbox{\tiny{R}}}_k\times\{0\}},
$$
that is $M^{\mbox{\tiny{R}}}_k\times\{0\}$ is collapsed to a single point which is the tip of the cone. We then define the min-max value
$$
	m=\inf_{f\in F} \sup_{\sigma\in\mathcal C} \mathcal E(f(\sigma)),
$$
where
$$
	F=\Bigr\{ f:\mathcal C\to H^m(M) \mbox{ continuous } \,:\, f(\sigma)=\varphi_{\lambda,\sigma}, \ \forall\sigma\in\partial\mathcal C=M^{\mbox{\tiny{R}}}_k \Bigr\},
$$
which is non-empty since $t\Phi\in F$. Still by Proposition \ref{prop-phi} we can take $\lambda\gg1$ sufficiently large such that
$$
	\sup_{\sigma\in\partial\mathcal C} \mathcal E(f(\sigma))= \sup_{\sigma\inM^{\mbox{\tiny{R}}}_k}\mathcal E(\varphi_{\lambda,\sigma})\leq -2L.
$$
On the other hand, we claim that
$$
	m\geq-L.
$$
To prove it, we just need to observe that $\partial\mathcal C=M^{\mbox{\tiny{R}}}_k$ is contractible in $\mathcal C$ (by construction of the cone) and thus $\Phi(M^{\mbox{\tiny{R}}}_k)$ is contractible in $f(\mathcal C)$ for any $f\in F$. Hence, we deduce by \eqref{non-contr} that $f(\mathcal C)$ can not be contained in $\mathcal E^{-L}$, which proves the claim.

\medskip

We conclude that the functional $\mathcal E$ has a min-max geometry at the level $m$ which in turn implies there exists a Palais-Smale sequence at this level.

\

\noindent \emph{Step 3.} Since the Palais-Smale condition is not available in this framework, we can not directly pass to the limit to obtain a critical point. To overcome this problem we use the so-called monotonicity trick jointly with the compactness property in Theorem \ref{th1.1}. This argument has been first introduced by Struwe in \cite{struwe} and has been then applied by many authors, see for example \cite{dm, ndiaye}. Therefore, we omit the details and just sketch the main ideas.

\smallskip

One considers a small perturbation $\mathcal E_\varepsilon$ of the functional so that the above min-max scheme applies uniformly. By using a monotonicity property of the perturbed min-max values $m_\varepsilon$ it is possible to obtain a bounded Palais-Smale sequence which then converges to a solution of the perturbed problem. We then pass to the limit as $\varepsilon\to0$ by using the compactness property in Theorem \ref{th1.1} to eventually recover a solution of the original problem. This concludes the proof.
\end{proof}

\medskip

Finally, we present the proof of the multiplicity result in Theorem \ref{th-mult}.
\begin{proof}[Proof of Theorem \ref{th-mult}]
Once the above analysis (needed to prove the existence result) is carried out, the multiplicity result is essentially a straightforward application of Morse inequalities. Thus, we will be sketchy and refer for example to \cite{bdm,de} for further details. Recall also that $\mathcal E$ is assumed to be a Morse functional. The (weak) Morse inequalities would assert that
\begin{align*}
	\#\{ \mbox{solutions of }\eqref{1.mliouville}\} &\geq \sum_{q\geq0}\#\{ \mbox{critical points of $\mathcal E$ in $\{-L\leq\mathcal E\leq L\}$ with index $q$}\}  \\
	& \geq \sum_{q\geq0} \mbox{{dim} } 	H_{q}(\mathcal E^L, \mathcal E^{-L}),
\end{align*}
where $H_{q}(\mathcal E^L, \mathcal E^{-L})$ stands for the relative homology group of $(\mathcal E^L, \mathcal E^{-L})$, see for example Theorem 2.4 in \cite{de}. Now, it is known that the high sublevels $\mathcal E^L$ are contractible. Roughly speaking, one can take $L\gg1$ sufficiently large so that there are no critical points above the level $L$ which then allows to construct a deformation retract of $\mathcal E^L$ onto $H^m(M)$, which is of course contractible, see for example the argument in \cite{mal}. Then, by the long exact sequence of the relative homology, it easily follows that
$$
H_{q}(\mathcal E^L, \mathcal E^{-L})\cong \widetilde H_q( \mathcal E^{-L}).
$$
But we already now from \eqref{inj} that
$$
H_*(M^{\mbox{\tiny{R}}}_k) \hookrightarrow H_*\bigr(\mathcal E^{-L}\bigr) \mbox{ injectively},
$$
thus
$$
\mbox{{dim} } \widetilde H_q( \mathcal E^{-L})\geq \mbox{{dim} } \widetilde H_q( M^{\mbox{\tiny{R}}}_k)
$$
and we are done.
\end{proof}

\

\begin{remark} \label{rem-general}
\emph{As already pointed out, for simplicity, all the argument has been carried out in the case $P_g^{2m}\geq0$. In general, one needs to modify the Adams-Trudinger-Moser inequality and its improvements by adding a bound to the component $u_{-}$ of the function $u$ lying in the direct sum of the negative eigenspaces of $P_g^{2m}$. As a consequence, in the low subleveles $\mathcal E^{-L}$ either the function $u$ concentrates or $u_{-}$ tends to infinity, or both alternative can hold. To express this alternative one can use the topological join between $M^{\mbox{\tiny{R}}}_k$ and a set representing the negative eigenvalues of $P_g^{2m}$. We refer the interested reader to \cite{dm}.}
\end{remark}

\medskip

\section{Appendix: Pohozaev identity}
Here we state a Pohozaev-type identity which is used in the blow-up argument.

\begin{lemma}
\label{lea.poho}
Let $B_r(0)$ be a ball in $\mathbb{R}^{2m}$, we have the following identities:
\begin{enumerate}
\item [(a).] If $m=2m_0,~m_0\geq1$, then
\begin{equation}
\label{a.poho1}
\begin{aligned}
&\int_{B_r(0)}x\cdot\nabla f(-\Delta)^{2m_0} fdx\\
&=\sum_{i=2}^{m_0}\int_{\partial B_r(0)}
2(i-1)\left((-\Delta)^{m-i}f\frac{\partial(-\Delta)^{i-1}f}{\partial\nu}-\frac{\partial(-\Delta)^{m-i}f}{\partial\nu}(-\Delta)^{i-1}f\right)ds\\
&\quad+\sum_{i=1}^{m_0}\int_{\partial B_r(0)}\left(
(-\Delta)^{m-i}f\partial_{\nu}\langle x,\nabla(-\Delta)^{i-1}f\rangle-\langle x,\nabla(-\Delta)^{i-1}f\rangle\frac{\partial(-\Delta)^{m-i}f}{\partial\nu}\right)ds\\
&\quad+\int_{\partial B_r(0)}\frac12|x|((-\Delta)^{m_0}f)^2ds.
\end{aligned}
\end{equation}

\item [(b).] If $m=2m_0+1,~m_0\geq1$, then
\begin{equation}
\label{a.poho2}
\begin{aligned}
&\int_{B_r(0)}x\cdot\nabla f(-\Delta)^{2m_0+1} f\\
&=\sum_{i=2}^{m_0}\int_{\partial B_r(0)}
2(i-1)\left((-\Delta)^{m-i}f\frac{\partial(-\Delta)^{i-1}f}{\partial\nu}-\frac{\partial(-\Delta)^{m-i}f}{\partial\nu}(-\Delta)^{i-1}f\right)ds\\
&\quad+\sum_{i=1}^{m_0}\int_{\partial B_r(0)}\left(
(-\Delta)^{m-i}f\partial_{\nu}\langle x,\nabla(-\Delta)^{i-1}f\rangle-\langle x,\nabla(-\Delta)^{i-1}f\rangle\frac{\partial(-\Delta)^{m-i}f}{\partial\nu}\right)ds\\
&\quad+\int_{\partial B_r(0)}\frac12|x|(\nabla(-\Delta)^{m_0}f)^2ds-2m_0\int_{\partial B_r(0)}\partial_{\nu}(-\Delta)^{m_0}f(-\Delta)^{m_0}fds\\
&\quad-\int_{\partial B_r(0)}\partial_{\nu}(-\Delta)^{m_0}f\langle x,\nabla(-\Delta)^{m_0}f\rangle ds.
\end{aligned}
\end{equation}
\end{enumerate}
Here $\nu$ is outward normal vector along the boundary $\partial B_r(0).$
\end{lemma}

\begin{proof}
The proof is based on the following identity
$$(-\Delta)\langle x,\nabla(-\Delta)^if\rangle
=2(-\Delta)f+\langle x,\nabla(-\Delta)^{i+1}f\rangle.$$
Using the second Green's identity repeatedly, we can get above formula by straightforward computations. We omit the details.
\end{proof}

\

\

\bibliographystyle{acm}

redpajama_set_name: RedPajamaArXiv


Semiconductor devices, from the earliest transistors to the projected spin-ristors, depend critically on defects. Methods to controllably add and minimize defects are essential for the success of optical, electronic, and spintronic based technologies, therefore, developing a detailed understanding of their fundamental physics and chemistry is mandatory for the creation of new game-changing devices.
This Gordon Research Conference is a unique biannual opportunity to discuss defects across all semiconductor materials and applications in an off-the-record format. By blending lecture hall discourse and social gatherings, young scientists and experienced professionals will learn the latest concerns and advances in semiconductors while enjoying the conference-friendly facilities and relaxing atmosphere of Colby-Sawyer College. The meeting will address the theoretical basics as well as the characterization and utility of point, line and extended defects in a broad range of topical materials. This includes defects in novel two-dimensional materials such as transition metal dichalcogenides and other graphene analogues along with semiconductors relevant to quantum emission processes. We also expect sessions to address wide bandgap nitride, oxide, and carbide semiconductors with evolving applications in electronics for power generation and lighting.
In addition to the GRC, graduate and post-doctoral students are encouraged to attend the pre-conference Gordon Research Seminar (GRS) designed by their peers. The two-day GRS acclimates young scientists to the unique style of Gordon Conferences and encourages them to share and present their ideas to their peers.
This GRC will be held in conjunction with the "Defects in Semiconductors (GRS)" Gordon Research Seminar (GRS). Those interested in attending both meetings must submit an application for the GRS in addition to an application for the GRC. Refer to the associated GRS program page for more information.
"Doping and Defects in β-Ga2O3"
"Defect Levels, Polaronic Distortions and Band Gaps"
"Spin-Dependent Recombination: How It Works and What We Learn"
"Liquid-Nitrogen-Temperature Photoluminescence Due to Radiation-Induced Defects for Quantification of Low-Level Carbon in Si"
"Scanning Tunneling Microscopy Characterization and Manipulation of Individual Defects in Semiconductors"
"Using Electron Paramagnetic Resonance to Identify and Characterize Native Defects and Impurities in β-Ga2O3 Crystals"
"Electrically Active Defects in β-Ga2O3"
"Quantum Emitters in Atomically Thin Materials"
"Vibrational Properties of Isolated Color Centers in Diamond"
"Elucidating the Role of Point Defects in the Performance of Thin-Film Solar Cells Through First-Principles Calculations"
"Structure and Vibrational Properties of Hydrogen-Related Native Defect Complexes in β-Ga2O3"
"Surface and Interfacial Defects in SrTiO3 Polar/Non-Polar Heterostructures"
"Influence of Solute Incorporation Mechanisms in Highly Mismatched Alloys"
"Investigating the Structural and Luminescence Properties of Semiconductors in the Scanning Electron Microscope"
"Donor Electron Spin Qubits in ZnO"
"3D Localized Excitons in Van der Waals Heterstructures"
"Defects in Metal Halide Perovskites"
"Origins of n-Type Doping Difficulties in Perovskite Stannates"
"Defect Engineering in Emerging Thermoelectric Materials"
"Defects in Si: What Is New?"
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Acquisition & Assistance under Award Number(s) DE-SC0018630.


A1 Excursion Adventure (P.) Ltd is a well known government licensed trekking and touring agency in Nepal. and accepts your bookings under the following terms and conditions.
We except Traveler's Cheque / Wire Transfer / Visa or Mater Card / Western Union / Cash - US Dollar - Euro - Pound - Japanese Yen - Hong Kong Dollar - Australian Dollar - Singapore Dollar - Etc.
Beneficiary: A1 Excursion Tours and Travels Pvt. Ltd.
Visa and Master cards are widely accepted in Nepal. So they are our preferred credit cards. American express, JCB and other major credit cards are also accepted but at times they are less convenient. Please check with us before payment.
Note that you are supposed to pay the bank charges (5% in Nepal) while making payments by credit cards by yourself.
If A1 Excursion Adventure (P.) Ltd. cancels your trip thanks to unforeseen circumstances like war, natural disasters, conflict, unfavorable climate etc., we would be happy to refer you alternative trip of a similar standard. Even if this doesn't fit your personal interest, all monies paid by you to our company will be refunded. But remember, any other expenses incurred by you as a result of this booking are entirely your responsibility.
A tour / trek guide who represents A1 Excursion Adventure deserves all authority during your tours. If you commit any unlawful act, you are compelled to leave the tour. No refund will be made in such case.
There for A1 Excursion Adventure Trek and its related organization shall not be responsible for any changes in the itinerary due to unavoidable circumstance such as govt. restriction, Land slide, road blockage, flood, snow political unrest, cancellation of flight, delay, sickness or accident, Any extra cost incurring there of shall be borne by the clients on the spots.


text:
A TCP server acting as a proxy to
[Californium](https://github.com/mkovatsc/Californium).
It's a companion to the node.js client available at
[californium-proxy-node](https://github.com/morkai/californium-proxy-node).

## Requirements

### Java

Version 1.6.0_37 or later available from [java.com](http://java.com/en/download/manual.jsp).

### [Naga - Simplified Java NIO asynchronous sockets](http://code.google.com/p/naga/)

Version 3.0 without EventMachine. JAR is included in the repository
for convenience (`lib/naga-no-em-3_0.jar`).

### [My fork](https://github.com/morkai/Californium) of the [Californium - CoAP framework in Java](https://github.com/mkovatsc/Californium)

JAR is included in the repository for convenience
(`lib/morkai-californium-0.8.4-SNAPSHOT.jar`).

## Usage

Build the JAR by executing the `californium-proxy-java/build`
script to create the `californium-proxy-java/run/cf-proxy.jar` file.

Run the server by executing:
```
cd califorium-proxy-java
java -cp run/cf-proxy.jar;lib/naga-no-em-3_0.jar;lib/morkai-californium-0.8.4-SNAPSHOT.jar CoapProxyServer
```
on Windows, or:
```
cd califorium-proxy-java
java -cp run/cf-proxy.jar:lib/naga-no-em-3_0.jar:lib/morkai-californium-0.8.4-SNAPSHOT.jar CoapProxyServer
```
on Linux.

Alternatively, use the bundled executable JAR:
```
java -jar califorium-proxy-java/run/cf-proxy-0.0.2.jar
```

By default, the TCP server will listen on port `1337` and output
all log messages. To change that, use the `--port` and `--log`
arguments.

The log argument expects a one of the [`java.util.logging.Level`](http://docs.oracle.com/javase/7/docs/api/java/util/logging/Level.html#field_summary)
values (case insensitive).
Use `OFF` to disable all messages.

For example, to silently listen on port `3000`:
```
java -jar califorium-proxy-java/run/cf-proxy-0.0.2.jar --port 3000 --log OFF
```

## Protocol

The server expects and sends packets of the following format:
```
<PACKET LENGTH><REQUEST ID><COAP FRAME>
```
Where:
  * `PACKET LENGTH` is a 32-bit unsigned integer and must be equal
    to a length of the `COAP FRAME+2`.
  * `REQUEST ID` is a 16-bit unsigned integer identifying this CoAP
    request (or a CoAP response to a request if send by the server).
    It's used by the client to easily match the responses to
    the requests.
  * `COAP FRAME` is a variable length, valid CoAP frame
    (request frame, if send by the client; response frame if send by
    the server).

See [Proxy.js](https://github.com/morkai/californium-proxy-node/blob/master/lib/Proxy.js)
from the [californium-proxy-node](https://github.com/morkai/californium-proxy-node)
project for an example implementation.

## License

This project is released under the
[MIT License](https://raw.github.com/morkai/californium-proxy-java/master/license.md).

redpajama_set_name: RedPajamaGithub


text:
Birdtown is a neighborhood near Cleveland, Ohio, United States.

Birdtown or Bird town may also refer to:

 Birdtown, Arkansas, a community in Arkansas, U.S.
 Bird Town, an American racehorse and broodmare
 "Birdtown," a municipal conservation program often organized by state Audubon societies
 Robbinsdale, Minnesota, given the nickname "Birdtown" by its residents

redpajama_set_name: RedPajamaWikipedia


text:
Mini-Series Masterpieces - DVD + Digital
Format DVD+D DVD
DVD+D DVD
$14.98 MSRP
Based on the Sequel to Margaret Mitchell's GONE WITH THE WIND by Alexandra Ripley
Scarlett, the six-hour mini-series based on Alexandra Ripley's best-selling sequel to Gone With the Wind, takes up precisely where that story left off, and sees Scarlett O'Hara off on a journey from Atlanta to Charleston and Savannah, to London, and to her ancestral home of Ballyhara in Ireland.
The motivating force behind the story is once again the turbulent romance between Scarlett and Rhett Butler. Separated from Scarlett and seeking a divorce, Rhett is now pursued passionately by his wife, who seeks a reconciliation. Frustrated in her effort and unwilling to win Rhett by telling him she is pregnant with their child, Scarlett leaves for Ireland, where she is caught up in the political upheaval of the day, and to England, where she must stand trial for a murder she did not commit.
Scarlett continues what is unarguably the most celebrated story of the American cinema, with all the richness of the original and with an epic sweep that today is seen only in long-form television.
Starring: Timothy Dalton, Joanne Whalley-Kilmer, Sean Bean, Stephen Collins, Annabeth Gish, Colm Meaney, Esther Rolle, Ann-Margret
Timothy Dalton, Joanne Whalley-Kilmer, Sean Bean, Stephen Collins, Annabeth Gish, Colm Meaney, Esther Rolle, Ann-Margret
Co-Starring:
Ann-Margret, Sean Bean, Jean Smart, George Grizzard
William Hanley, Alexandra Ripley
Tech Specs:
Discs:
Region Code:
Street Date:
Running The Bases
Ultraman Max – The Complete Series
Ultraman Kids 3000: The Complete Series
30 Rock - The Complete Series

redpajama_set_name: RedPajamaCommonCrawl


For the holidays, Jordan Brand is aiming to drop a collection, which will be including many silhouettes. Occasionally, the Air Jordan 3 Mocha is coming back this year 2018, after its las seen in 2001. This pair will come in its original white color with Mocha. Much like the 2001 pair, it features a white leather upper, while Dark-Mocha elephant print overlays the pair. A matching Dark Mocha rubber outsole finished off this design.
You will be able to enjoy this Air Jordan 3 " Mocha" 2018 is set to drop on December 15th at select Jordan Brand retailers. Check out more detailed images below, be sure to grab a pair. The retail price tag is set $190 USD.


text:
La Cage Aux Folles a polished Broadway in Chicago gem
Broadway In Chicago La Cage Aux Folles:
"I Am What I Am," A Polished Broadway Gem
Story by Dwight Casimere
Photos by Paul Kolnikstud for Broadway In Chicago
-George Hamilton as Georges with "Les Cagelles"
-George Hamilton (Georges) and Christopher Sieber (Albin) in the finale
Broadway in Chicago's current production of La Cage Aux Folles has the look and feel of a polished Broadway show, and with good reason. The majority of the cast are veterans of at least one of its several Broadway incarnations, including one of its stars, Christopher Sieber, who plays the centerpiece character, Albin, the self-described "one transvestite homosexual," who provides the dramatic fulcrum for the musical. Indeed, it is Sieber, in the person of Albin, who delivers the show-stopping anthem, "I Am What I Am" at the end of Act 1, which serves as the emotional heartbeat of La Cage Aux Folles, at Chicago's Bank of America Theatre, through Jan. 1, 2012.
La Cage aux Folles is the only Broadway musical that has won the Tony Award for Best Revival of a Musical twice, and the only show that has won a Best Production Tony Award (Best Musical or Best Revival of a Musical) for each of its Broadway productions.
Christopher Sieber starred in La Cage on Broadway as the character Georges, who is played in Chicago by the venerable actor George Hamilton.
A 50 -year veteran of film and television, Hamilton, is a living monument to his profession. He plays Albin's "male" partner, with dignity and aplomb. In addition to his encyclopedic film and television credits, he has distinguished himself on Broadway, playing the starring role of Billy Flynn three times in the smash hit Chicago. There are key moments in La Cage that offer a window into the inner genius of his skills as a performer. Hamilton employs a dramatic pause in several scenes that allows his character's 'voice' to speak volumes. He achieves in silence, an eloquence equal to the words of Harvey Fierstein's often poignant script.
Georges (Hamilton) is the proprietor of the French Riviera nightclub of the title, which is the setting for the racy transvestite revue that stars his life-partner, Albin (Sieber). The two have raised a son, Jean-Michel (Billy Harrigan Tighe), whom Georges sired in a drunken one-night stand with a chorus girl at the Lido in Paris, where Georges once starred. Georges and Albin's sequin-covered world is suddenly rent asunder when Jean-Michel announces that he is about to be married and wants to bring his fiancé, Anne (an ebullient Doris-Day-ish Allison Blair McDowell) and her arch-conservative parents, the Dindons (an irascible Bernard Burak Sheredy and a comically inept Cathy Newman) home.
M. Dindon is a right-wing politician, running for office on a platform based on his crusade against the libertine elements that run rampant in the south of France. His avowed mission is to eradicate transvestite clubs and their ilk from the face of the earth. With a substantial dowry at risk, Jean-Michel wants Albin gone from the premises, so that he can bring the girl's parents into a seemingly 'normal' home. (Albin later delivers one of the great, standout lines of the entire evening, when he tongue-in-cheek (pardon the pun) mangles Dindon's name, calling him "Ding-dong.")
Albin, who raised Jean-Michel, is understandably crushed. Hamilton reprimands his son and shames him through the moving song "Look Over There."
Georges attempts to smooth things over with Albin over a rare lunch at an outdoor café on the Promenade, and sings the lovely "Song on the Sand." It's hard not to hear the great Gene Barry's voice in the back of your mind on that one, but Hamilton comports himself ably. Although his voice wavers at time, he sings with conviction and delivers the lyric with authority. He's a master at 'selling' a song, as evidenced in the comic scene that incorporates the song, "Masculinity," in which Georges attempts to teach the effeminate Albin to adopt the mannerisms and ungainly walk of John Wayne, with hilarious results!
The club's male chorus dancers, "Les Cagelles" are a consistent delight throughout the performance. We hear them singing, in cabaret cadence, "I Am What I Am, and what I Am is an illusion," although we can see the pecs and biceps under all the pasties and sequins and a distinct "package" under the g-strings. Their athleticism particularly comes into focus in the "Birdcage" dance that opens the second act. Jeigh Madjus as the "butler (maid)" Jacob is a sassy bundle of comic energy throughout.
If you've tired of the marathon of Holiday treacle and commercialism, La Cage Aux Folles is a welcome tonic, which treats a sensitive subject with humanity and humor. It gives the Holiday chestnut "Peace on Earth, Goodwill toward Men" a deeper and more universal meaning.
Broadway In Chicago: The Addams Family up to old w...
Robert Joffrey's Nutcracker celebrates 25 years of...
Met Live HD: Gounod's Faust a "Magnificent ride"
Music of Italian Innovators sparkles during Art Ba...
86 year-old drummer Roy Haynes "Snap, Crackles" at...
Met Live HD Rodelinda a showcase for great singers


That mani-pedi can pose a health risk for both people involved. Possible fungus or bacteria for the salon client. Possible ill effects from fumes for the salon worker.
The standards are designed to protect the health of nail technicians, clients, and visitors from the risk of injury or infection due to unsanitary conditions and exposure to hazardous chemicals….
Last month the board gave preliminary approval to a proposed regulation that would standardize nail salon practices, such as hand washing, sterilization of tools, and labeling of chemicals. The proposed regulation comes amid growing concern about the health and safety of workers and clients. Nail technicians, who are exposed to a myriad of potentially hazardous chemicals every day, may experience headaches, dizziness, fatigue, breathing difficulties, and infertility. Inadequate sterilization can lead to allergic reactions, bacterial and fungal infections, and injuries.
The proposed regulation would require that the Boston Public Health Commission inspect and permit all nail salons in Boston. An owner of a nail salon found to be in violation of any provision of the proposed regulation may be fined $100 for the first violation, $200 for the second violation within a 12-month period, and $300 for the third and all subsequent violation(s) within a 12- month period. Repeated violations may result in suspension or revocation of the salon's permit to operate.
The Massachusetts Department of Public Health currently regulates nail salons statewide. The proposed regulations would allow the Boston Public Health Commission to, for the first time, inspect and permit them nail salons, according to a Commission spokeswoman.


text:
Q: php mkdir, folder name and structure I need to make an new folder on my server, and this also works just fine using mkdir()
But can someone tell me how to make both folder name and structure using mkdir()??
Folder name will be = $_SESSION['username'];
and structure inside the new folder should be = /folder1/folder2/folder3/folder4/
Thanks for your time.
/thar

A: You need to set third argument to true which allows you creation of nested directories:
mkdir($_SESSION['username'] . '/folder1/folder2/folder3/folder4/', 0777, true);

redpajama_set_name: RedPajamaStackExchange


Studying in the United States is one of the biggest dreams for many prospective students around the world. Academic excellence, research and career opportunities, the American college culture or athletic scholarships are only some of the many compelling reasons to move to the U.S. for a college degree.
However, international students often have to overcome barriers such incredibly high tuition fees and cost of living, challenging application programs or visa-related issues.
It's no surprise that international students from all over the world choose to study here every year. In fact, there were over 1 million international students in the U.S. in 2017/2018, which was 5.5% of the total population of those enrolled in higher education in the U.S.
Aside from rank, there are two other factors that play a major role in helping international students decide to study in the U.S.: cost (tuition assistance is often harder to secure for international students) and size of the existing international student population, which usually goes hand-in-hand with current resources available to international students.
We've put together a list of the top 20 U.S. universities by rank that fall in to at least two of the three of the previously mentioned categories to form our list of best U.S. universities for international students.
Princeton is currently ranked as the top school in the U.S. according to the U.S. News and World Report. Its Davis International Center offers programs to help its international students adjust to life in the U.S., the most of which hail from China, Canada, and India.
Often topping university ranking lists (its World University rank is #3), Harvard is a dream school for many foreign students. In fact, its international student population has steadily increased over time, with 103 countries represented in its undergraduate population alone. Its proximity to other universities in the greater Boston area is widely appealing to international students looking to make connections beyond the Harvard campus.
Located in one of the most diverse cities in the U.S., Columbia's total student population hails from 151 countries, and over 14,000 of its nearly 33,000 students are from outside of the U.S. In fact, Columbia is the Ivy League university with the most international students.
You can use your international credit report through Nova Credit to apply for credit cards, apartments, loans and more in the United States.
Students flock from all over the world to study at MIT, which is currently ranked as the #1 World University and is well-known for its computer science program. The school maintains a strong global presence, with centers and campuses in Abu Dhabi, Singapore, and China. The majority of MIT's international students are from China, Canada, India, South Korea, and France, among many others.
Yale students come from 118 countries, and its Office of International Students & Scholars assists students with issues surrounding cultural adjustment, visas, and employment. Yale is a popular choice for international students who are seeking financial aid, as the university has a need-blind policy for all applicants.
Stanford's Bechtel International Center helps students from across the globe—including the ones hailing from 63 countries from its Class of 2022—adjust to life in the United States.
Duke's international student population has grown substantially over the years. In 2017, the university reported that it saw a 42% increase in international students over the last 5 years, with 60% of those students hailing from China, India, South Korea, Canada, and Turkey.
16% of UPenn's Class of 2022 is made up of international students, with 398 students coming from 87 different countries.
Johns Hopkins is home to over 4,000 international students from over 130 countries—its undergraduate population alone represents 57 nations.
Caltech's student body is largely made up of international students, the majority of whom come from China, India, and Canada. Given its research university status, it's a popular option for those who are pursuing doctorates.
Dartmouth offers an extensive number of resources for its international students to ensure that they're set up for success, including international student profiles for its prospective applicants.
In May of 2017, Brown University established the Global Brown Center for International Students to foster intercultural dialogue between students. Its Class of 2022 is comprised of students from 76 nations, including China, India, the United Kingdom, Canada and Singapore.
According to its website, Vanderbilt has enrolled students from other countries for all but two years of its existence (it was founded in 1875), and has had students that come from all over the world, including Germany and Mexico.
With seven colleges offering disciplines like hotel administration and agricultural sciences, over 5,000 students from all over the world comprise Cornell's international population. Aside from China, India, Canada, South Korea, and Taiwan, Cornell enrolls a large population of students from Hong Kong, Singapore, Mexico, Turkey, and the United Kingdom.
Nova helps you relocate to the U.S. with excellent guides and tips for you if you're considering a move, or have recently arrived. Stay up-to-date, unsubscribe anytime.
In the last decade, the international student population at Rice has increased by 59%. In fact, nearly one in every four students at Rice is international. While 80% of its international population is from Asia, Rice has also attracted students from countries such as Australia, Nigeria, Belarus, and Mexico, which borders Rice's home state of Texas.
Widely regarded as one of the top state universities in the U.S., UCLA is home to nearly 7,000 international students from over 100 countries, including Japan, Australia, and Brazil.
Emory, located in sunny Georgia, has become a popular choice for international students. Its international student population increased by 35% between 2013 and 2017, with its College of Arts & Sciences, Laney Graduate School, and Goizueta Business School hosting the largest number of international students.
Nestled in the heart of the American capital of Washington, D.C., Georgetown is highly regarded for its international studies programs. It's no wonder, then, that Georgetown is one of the best universities for international students. Countries represented include Indonesia, Kosovo, and the Philippines.
Berkeley is often named the #1 public university worldwide and is continually ranked as a top state university in California. Its international student body has increased rapidly over the past two years, especially for its graduate programs.
USC is home to over 11,000 international students, the majority of whom come from China, India, South Korea, and Taiwan.
Of course, there are many other American colleges and universities that are not on this list but are still great options for international students. It's also important to keep in mind that the majority of U.S. universities have specific requirements for international applications, including proof of English language proficiency and credential evaluations, so set aside some time to do some research before determining the best U.S. universities for you to consider.


Posted October 4th, 2017 by Kim Higdon & filed under Adhesive Tapes, Converting Services.
Save time and money on the assembly line by outsourcing your manufacturing process with a custom tape converter. No matter what products you are trying to develop, there is a custom adhesive tape solution from a converter that can provide materials on time and in quantities needed for your production. Specialty adhesives and tapes can combine raw or semi-finished materials to fabricate finished adhesive and non-adhesive parts that are used during the manufacturing of finished goods. Custom tapes can be used with a variety of materials including cloths, foams, plastics, rubbers, polyesters, adhesives, silicones, metals, and more.
Additionally, by partnering with a dual-house tape converter, you can rely on their knowledge of materials and technical expertise to save you time and money. A dual-house converter can assist with in-house design, rapid prototyping, testing, mass production, and cost control. They are able to assist your engineers and manufacturers in a constant effort to maximize efficiency.
Experience: Do they have the experience and knowledge about creating custom adhesive tape products for your industry?
Capabilities: Do they have the design, material, facility, and production capabilities to complete your project?
Quality and Consistency: Do they have a proven track record of delivering quality products?
Customer Service: Do they respond in a timely manner and resolve issues as they arise?
Competitive Pricing: Do they think in terms of value or price?
We have a team of engineers that are able to make your designs ready for production and ensure that your project will work the first time. We utilize tooling design to reduce scrap, increase yield, and achieve your design goals. Additionally, our equipment gives you access to custom tape solutions with no capital costs passed onto you and allows us to provide you with the dimensions you need.
Can-Do National Tape is the global leader in the Dual-House Tape Converting industry and is ISO certified. Our sales force and management has over 100 years of combined experience and continues to work with a variety of innovative materials. The company's converting expertise includes capabilities for a wide variety of pressure sensitive adhesives, non-adhesives, and heat activated thermoplastics. Can-Do's resources are worldwide, but are as close as your door. We believe customer service and technical expertise are fundamental in establishing healthy long-term business relationships. Give us a call to see how Can-Do National Tape can convert your ideas into reality.


text:
::::: : the wood : davidrobins.net
Looking for meaning
News, Baby ·Tuesday January 15, 2019 @ 19:19 EST (link)
I ended my last note by saying I miss my family. I miss my son especially; he had no choice in being kidnapped and removed from his father who loves him. And I wonder why Honey did it. It's easy to say "she's evil"—her acts certainly have been, and cruel too, in holding out hope when she was merely biding her time to do the most harm she could with legal safety, but that's not explanatory. What would motivate her to such a thing? Clearly she was unhappy; that's built-in, you don't leave a situation where you're happy. But in the eight months I drove there every other week trying to get her to come back, to restore our family rather than tear it apart, she was cold, she had thrown up a wall between us; it is evident now that from the very beginning she intended to go through with this evil and had no intention whatsoever of turning from it.
Had she been unfaithful (more than in the sense of running off and not keeping the promises and vows she made) and turned to someone else? I wouldn't have thought it, but I wouldn't have thought she could run off and be as cruel as she had been, to keep my son from me and lie to me day after day. She had opportunity with all of her visits to WV and all the time she spent online while I was at work.
Was she just pining for WV, and wanting to give up on marriage and instead regress to a comfortable environment where she could pretend to be a child again, taken care of by family—instead of by me—playing where she grew up, hanging out with her former social groups, going to her old church, which for her and her family has always been about socializing rather than about God? That's a possibility too, I suppose. A few advisers have independently come up with the idea that she never was a Christian, she had no "fruit" of a Christian life; and Paul says to let the unbelieving spouse depart and frees the one left to remarry.
Or perhaps she's not rational. One idea I've also been shown is a postpartum condition (which extends beyond just the oft-heard "postpartum depression" into a variety of issues), for which she should be evaluated if it is possible. Situations exactly like this, where a wife does a complete turn in her attitudes toward a kind and loving husband and runs off with a young child have been described online by several people across different sites. Such conditions can be treated with medicine and psychology, and denial of such a condition is also common.
Would I take her back? I've been asked that a lot too. I loved, trusted, and cared for her more than anyone ever, so very much, she was everything to me; and of course trusting like that allows for hurt like this. But yes, I would. We would need to work through whatever the issues were with counseling, for a long time I expect, but I would do whatever was necessary to put our family back together.


text:
POSTED | October 11, 2020
Board welcomes new directors for 2020
The Friends of Presqu'ile Park recently held elections for the Board of Directors. Four members of the Board are continuing into a subsequent year in their terms. They are: Linda Alkenbrack, Doreen Cable, Donal Gray and Larry Paradis.
Two new Directors were elected at the AGM; they are Christine McIvor and Wayne Meyers. Also, former Director Phil Mcrae has returned to the board for another term after a year off. Read more about these Directors below:
Christine McIvor
Chris is the new Board Chair for the Friends. She brings extensive experience in fund-raising and managing non-profit organizations to the organization. She has worked in Arts Administration for a Symphony association, the National ballet School and Arts Etobicoke. She currently serves as a Returning Officer for Elections Ontario.
Chris has served on the Board of Directors of Theatre Ontario and the Canadian Parks and Recreation Association. Her first involvement with the friends was as a volunteer at Christmas @ Presqu'ile.
Chris resides in Quinte West with her husband Bjorn. They are avid campers, day trippers and award-winning amateur photographers.
Phil McRae
Phil is returning to the Board of the Friends to act as the Chair of Environmental Initiatives, a post he formerly held for six years.
Phil resides in Oshawa; but is a life-long cottager at Presqu'ile. He is retired from a long career with General Motors. He is also a member of the first graduating class from Trent University.
In addition to his long volunteer service to the Friends, Phil is a former director of the Oshawa Kinsman Club. He currently serves as the leader of SOS (seniors on Spokes) Bicycle Group.
Welcome back, Phil!
Wayne Myers
Wayne has over 35 years of experience in the finance field. He served as Comptroller for a large trucking firm in the Trenton area.
Wayne has volunteered at Christmas @ Presqu'ile and also assisted the Friends' treasurer with banking duties. Wayne will serve as a Director-at-large.
Wayne is a life- long Presqu'ile Park visitor. He and his wife and family are regular campers at the Park, often spending most of the summer.


Telect offers a family of heavy duty shelves for our Datacenter family of racks, with a maximum weight rating of 400lbs. The shelves are available for either a 19" mount or a 23" wide mount, and have two adjustable widths. The shelves fits either a 24" to 30" deep rack or a 36"-42" deep rack. You can use the following table to find the heavy duty shelf that best meets your application.


Gillan Anderson scrubs up very nicely for important events. David Duchovny does not bother to scrub up at all for important events. The End.
This entry was posted on July 26, 2008 at 10:31 pm and is filed under North Americana, The X-Files.	You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.


When everything has gone soft and mooshy, add the Chicken Stock and boil it for about 15 minutes.
Let it cool down a bit and then pour it into a food processor [or you could use a Bamix/Stick Blender] and puree it until it has a fine texture.
Back into the saucepan, add the cream and gently simmer for 10x minutes, but make sure it doesn't boil.
When the bread is nicely toasted with the grill lines criss-crossed [because that looks really cool], rub garlic onto both side of the bread and add a light "dusting" of ground black pepper – but not too much!