text:
\section{Introduction}

\noindent
Recently, considerable attention has been given to fractional Sobolev Spaces and corresponding nonlocal equations, in particular to the ones driven by the fractional powers of the Laplacian.
In fact, this operator naturally arises in several
areas of research and find applications in optimization, finance, the thin obstacle problem, phase transitions, anomalous diffusion, crystal dislocation, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows and water waves. For more details and applications see \cite{Applebaum}, \cite{BKW}, \cite{Cabsolmor}, \cite{CV}, \cite{CT}, \cite{DL}, \cite{LY2}, \cite{Silvestre}, \cite{SirVal}, \cite{Stoker}, \cite{Toland} and references therein.

\noindent
The purpose of the present paper is to study the $T$-periodic solutions to the problem
\begin{equation}\label{P}
\left\{
\begin{array}{ll}
[(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{ in } (0,T)^{N}   \\
u(x+Te_{i})=u(x)    &\mbox{ for all } x \in \mathbb{R}^{N}, \quad i=1, \dots, N
\end{array},
\right.
\end{equation}
where $s\in (0,1)$, $N >2s$, $(e_{i})$ is the canonical basis in $\mathbb{R}^{N}$ and $f:\mathbb{R}^{N+1}\rightarrow \mathbb{R}$ is a function satisfying the following hypotheses:
\begin{compactenum}[($f1$)]
\item $f(x,t)$ is $T$-periodic in $x \in \mathbb{R}^{N}$; that is $f(x+Te_{i},t)=f(x,t)$;
\item $f$ is continuous in $\mathbb{R}^{N+1}$;
\item $f(x,t)=o(t)$ as $t \rightarrow 0$ uniformly in $x\in \mathbb{R}^{N}$;
\item there exist $1<p<2^{\sharp}_{s}-1=\frac{2N}{N-2s}-1$ and $C>0$ such that
$$
|f(x,t)|\leq C(1+|t|^{p})
$$
for any $x\in \mathbb{R}^{N}$ and $t\in \mathbb{R}$;
\item there exist $\mu>2$ and $r_{0}>0$ such that
$$
0<\mu F(x,t)\leq t f(x,t)
$$
for $x\in \mathbb{R}^{N}$ and $|t|\geq r_{0}$. Here $\displaystyle{F(x,t)=\int_{0}^{t} f(x,\tau) d\tau}$;
\item $tf(x,t) \geq 0$ for all $x\in \mathbb{R}^{N}$ and $t\in \mathbb{R}$.
\end{compactenum}
We notice that $(f2)$ and $(f5)$ imply the existence of two constants $a,b>0$ such that
\begin{equation*}
F(x,t) \geq a|t|^{\mu}-b \mbox{ for all } x\in \mathbb{R}^{N}, t\in \mathbb{R}.
\end{equation*}
Then, since $\mu>2$, $F(x,t)$ grows at a superquadratic rate and by $(f5)$, $f(x,t)$ grows at a superlinear rate as $|t| \rightarrow \infty$. \\
Here, the operator $(-\Delta_{x}+m^{2})^{s}$ is defined through the spectral decomposition, by using the powers of the eigenvalues of $-\Delta+m^{2}$ with periodic boundary conditions.\\
Let $u\in\mathcal{C}^{\infty}_{T}(\mathbb{R}^{N})$, that is $u$ is infinitely differentiable in $\mathbb{R}^{N}$ and $T$-periodic in each variable.
Then $u$ has a Fourier series expansion:
$$
u(x)=\sum_{k\in \mathbb{Z}^{N}} c_{k} \frac{e^{\imath \omega k\cdot x}}{{\sqrt{T^{N}}}} \quad (x\in \mathbb{R}^{N})
$$
where
$$
 \omega=\frac{2\pi}{T}\mbox{ and } \; c_{k}=\frac{1}{\sqrt{T^{N}}} \int_{(0,T)^{N}} u(x)e^{- \imath \omega k \cdot x}dx \quad (k\in \mathbb{Z}^{N})
$$
are the Fourier coefficients of $u$.
The operator $(-\Delta_{x}+m^{2})^{s}$ is defined by setting
\begin{equation*}\label{nfrls}
(-\Delta_{x}+m^{2})^{s} \,u=\sum_{k\in \mathbb{Z}^{N}} c_{k} (\omega^{2}|k|^{2}+m^{2})^{s} \, \frac{e^{\imath \omega k\cdot x}}{{\sqrt{T^{N}}}}.
\end{equation*}
\noindent
For  $\displaystyle{u=\sum_{k\in \mathbb{Z}^{N}} c_{k} \frac{e^{\imath \omega k\cdot x}}{{\sqrt{T^{N}}}}}$ and $\displaystyle{v=\sum_{k\in \mathbb{Z}^{N}} d_{k} \frac{e^{\imath \omega k\cdot x}}{{\sqrt{T^{N}}}}}$, we have that
$$
\mathcal{Q}(u,v)=\sum_{k\in \mathbb{Z}^{N}} (\omega^{2}|k|^{2}+m^{2})^{s} c_{k} \bar{d}_{k}
$$
can be extended by density to a quadratic form on the Hilbert space
$$
\mathbb{H}^{s}_{m,T}=\Bigl\{u=\sum_{k\in \mathbb{Z}^{N}} c_{k} \frac{e^{\imath \omega k\cdot x}}{{\sqrt{T^{N}}}}\in L^{2}(0,T)^{N}: \sum_{k\in \mathbb{Z}^{N}} (\omega^{2}|k|^{2}+m^{2})^{s} \, |c_{k}|^{2}<\infty \Bigr\}
$$
endowed with the norm
$$
|u|_{\mathbb{H}^{s}_{m,T}}^{2}=\sum_{k\in \mathbb{Z}^{N}} (\omega^{2}|k|^{2}+m^{2})^{s} |c_{k}|^{2}.
$$
When $m=1$ we set $\mathbb{H}^{s}_{T}=\mathbb{H}^{s}_{1,T}$. \\
In $\mathbb{R}^{N}$, the physical interest of the non-local operator $(-\Delta+m^{2})^{s}$ is manifest in the case $s=1/2$: it is the Hamiltonian for a (free) relativistic particle of mass $m$; see for instance \cite{A1}, \cite{FL}, \cite{LL}, \cite{LY1}, \cite{LY2}.
In particular, such operator is deeply connected with the Stochastic Process Theory: in fact it is an infinitesimal generator of a Levy process called $\alpha$-stable process; see \cite{Applebaum}, \cite{CMS} and \cite{Ryz}.\\
Problems similar to (\ref{P}) have been also studied in the local setting.
The typical example is given by
\begin{equation}\label{P''}
\left\{
\begin{array}{ll}
Lu=f(x,u) &\mbox{ in } \Omega   \\
u=0   &\mbox{ on } \partial \Omega
\end{array},
\right.
\end{equation}
where $L$ is uniformly elliptic, $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ and $f(x,t)$ is a continuous function satisfying the assumptions $(f3)$-$(f5)$. It is well known that (\ref{P''}) possesses a weak solution which can be obtained as critical point of a corresponding functional by means of minimax methods; see for instance \cite{AR}, \cite{MawWill}, \cite{Rab}, \cite{Struwe} and \cite{Willem}.
\smallskip

\noindent
The aim of the following paper is to study (\ref{P''}) in the periodic setting, when we replace
$L$ by $(-\Delta+m^{2})^{s}-m^{2s}$, $m\geq 0$ and $s\in (0,1)$.
We remark that the problem $(\ref{P})$ with $s=\frac{1}{2}$ has been investigated by the same author in \cite{A2}. In this paper, we extend the results in \cite{A2} to the more general operator $(-\Delta+m^{2})^{s}-m^{2s}$, with $s\in (0,1)$.
\medskip

\noindent
Our first result is the following
\begin{thm}\label{thm1}
Let $m>0$ and $f:\mathbb{R}^{N+1} \rightarrow \mathbb{R}$ be a function satisfying the assumptions $(f1)-(f6)$.
Then there exists a solution $u\in \mathbb{H}^{s}_{m,T}$ to (\ref{P}). In particular, $u$ belongs to $\mathcal{C}^{0,\alpha}([0,T]^{N})$ for some $\alpha \in (0,1)$.
\end{thm}

\noindent
To study the problem $(\ref{P})$ we will give an alternative formulation of the operator $(-\Delta+m^{2})^{s}$ with periodic boundary condition which consists to realize it as an operator that maps a Dirichlet boundary condition to a Neumann-type condition via an extension problem on the half-cylinder $(0,T)^{N} \times (0,\infty)$; see \cite{A2} for the case $s= \frac{1}{2}$.
We recall that this argument is an adaptation of the idea originally introduced in \cite{CafSil} to study the fractional Laplacian in $\mathbb{R}^{N}$ (see also \cite{CabSir1, CabSir2}) and subsequently generalized for the case of the fractional Laplacian on bounded domain (see \cite{CabTan,  CapDavDupSir}).

\noindent
As explained in more detail in Section $3$ below, for $u\in \mathbb{H}^{s}_{m,T}$ one considers the problem
\begin{equation*}
\left\{
\begin{array}{ll}
-\dive(y^{1-2s} \nabla v)+m^{2}y^{1-2s}v =0 &\mbox{ in }\mathcal{S}_{T}:=(0,T)^{N} \times (0,\infty)  \\
v_{| {\{x_{i}=0\}}}= v_{| {\{x_{i}=T\}}} & \mbox{ on } \partial_{L}\mathcal{S}_{T}:=\partial (0,T)^{N} \times [0,\infty) \\
v(x,0)=u(x)  &\mbox{ on }\partial^{0}\mathcal{S}_{T}:=(0,T)^{N} \times \{0\}
\end{array},
\right.
\end{equation*}
from where the operator $(-\Delta_{x}+m^{2})^{s}$ is obtained as
$$
-\lim_{y\rightarrow 0} y^{1-2s} \frac{\partial v}{\partial y}(x,y) = \kappa_{s} (-\Delta_{x} + m^{2})^{s} u(x)
$$
in weak sense and $\displaystyle{\kappa_{s}= 2^{1-2s} \frac{\Gamma(1-s)}{\Gamma(s)}}$.

\noindent
Thus, in order to study (\ref{P}), we will exploit this fact to investigate the following problem
\begin{equation}\label{R}
\left\{
\begin{array}{ll}
-\dive(y^{1-2s} \nabla v)+m^{2}y^{1-2s}v =0 &\mbox{ in }\mathcal{S}_{T}:=(0,T)^{N} \times (0,\infty)  \\
v_{| {\{x_{i}=0\}}}= v_{| {\{x_{i}=T\}}} & \mbox{ on } \partial_{L}\mathcal{S}_{T}:=\partial (0,T)^{N} \times [0,\infty) \\
\frac{\partial v}{\partial \nu^{1-2s}}=\kappa_{s} [m^{2s}v+f(x,v)]   &\mbox{ on }\partial^{0}\mathcal{S}_{T}:=(0,T)^{N} \times \{0\}
\end{array},
\right.
\end{equation}
where
$$
\frac{\partial v}{\partial \nu^{1-2s}}:=-\lim_{y \rightarrow 0} y^{1-2s} \frac{\partial v}{\partial y}(x,y)
$$
is the conormal exterior derivative of $v$.\\
The solutions to $(\ref{R})$ are obtained as critical points of the functional $\mathcal{J}_{m}$ associated to (\ref{P})
 $$
 \mathcal{J}_{m}(v)=\frac{1}{2} ||v||_{\mathbb{X}_{m,T}^{s}}^{2}-\frac{m^{2s}\kappa_{s}}{2}|v(\cdot,0)|_{L^{2}(0,T)^{N}}^{2} -\kappa_{s}\int_{\partial^{0}\mathcal{S}_{T}} F(x,v) \,dx
 $$
defined on the space $\mathbb{X}_{m,T}^{s}$, which is the closure of the set of smooth and $T$-periodic (in $x$) functions in $\mathbb{R}^{N+1}_{+}$ with respect to the norm
$$
 ||v||_{\mathbb{X}_{m,T}^{s}}^{2}:=\iint_{\mathcal{S}_{T}} y^{1-2s} (|\nabla v|^{2}+m^{2s} v^{2}) dx dy.
$$
More precisely, we will prove that, for any fixed $m>0$, $\mathcal{J}_{m}$ satisfies the hypotheses of the Linking Theorem due to Rabinowitz \cite{Rab}.
\medskip

\noindent
When $m$ is sufficiently small, we are able to obtain estimates on critical levels $\alpha_{m}$ of the functionals $\mathcal{J}_{m}$ independently of $m$.
In this way,  we can  pass to the limit as $m\rightarrow 0$ in (\ref{R}) and we deduce the existence of a nontrivial solution to the problem
\begin{equation}\label{P'}
\left\{
\begin{array}{ll}
(-\Delta_{x})^{s} u=f(x,u) &\mbox{ in } (0,T)^{N}   \\
u(x+Te_{i})=u(x)    &\mbox{ for all } x \in \mathbb{R}^{N}, \quad i=1, \dots, N
\end{array}.
\right.
\end{equation}
This result can be stated as follows
\begin{thm}\label{thmdue}
Under the same assumptions on $f$ of Theorem \ref{thm1},
the problem (\ref{P'}) admits a nontrivial solution $u\in \mathbb{H}^{s}_{T}\cap \mathcal{C}^{0,\alpha}([0,T]^{N})$.
\end{thm}

\noindent
The paper is organized as follows: in Section $2$ we collect some preliminaries results which we will use later to study the problem $(\ref{P})$; in Section $3$ we show that the problem (\ref{P}) can be realized in a local manner through the nonlinear problem (\ref{R});
in Section $4$ we verify that, for any fixed $m>0$, the functional $\mathcal{J}_{m}$ satisfies the Linking hypotheses; in Section $5$ we study the regularity of solutions of problem $(\ref{P})$; in the last Section we show that we can find a nontrivial H\"older continuous solution to $(\ref{P'})$ by passing to the limit in $(\ref{P})$ as $m\rightarrow 0$.

\section{Preliminaries}

In this section we introduce some notations and facts which will be frequently used in the sequel of paper.

We denote the upper half-space in $\mathbb{R}^{N+1}$ by
$$
\mathbb{R}^{N+1}_{+}=\{(x,y)\in \mathbb{R}^{N+1}: x\in \mathbb{R}^{N}, y>0 \}.
$$
Let $\mathcal{S}_{T}=(0,T)^{N}\times(0,\infty)$ be the half-cylinder in $\mathbb{R}^{N+1}_{+}$ with basis $\partial^{0}\mathcal{S}_{T}=(0,T)^{N}\times \{0\}$
and we denote by $\partial_{L}\mathcal{S}_{T}=\partial (0,T)^{N}\times [0,+\infty)$ its lateral boundary.

With $||v||_{L^{r}(\mathcal{S}_{T})}$ we always denote the norm of $v\in L^{r}(\mathcal{S}_{T})$ and with $|u|_{L^{r}(0,T)^{N}}$ the $L^{r}(0,T)^{N}$ norm of $u \in L^{r}(0,T)^{N}$.

Let $s\in (0,1)$ and $m> 0$. Let $A\subset \mathbb{R}^{N}$ be a domain. We denote by $L^{2}(A\times \mathbb{R}_{+},y^{1-2s})$
the space of all measurable functions $v$ defined on $A\times \mathbb{R}_{+}$ such that
$$
\iint_{A\times \mathbb{R}_{+}} y^{1-2s} v^{2} dxdy<\infty.
$$
We say that  $v\in H^{1}_{m}(A\times \mathbb{R}_{+},y^{1-2s})$ if $v$ and its weak gradient $\nabla v$ belong to $L^{2}(A\times \mathbb{R}_{+},y^{1-2s})$.
The norm of $v$ in $H^{1}_{m}(A\times \mathbb{R}_{+},y^{1-2s})$ is given by
$$
\iint_{A\times \mathbb{R}_{+}} y^{1-2s} (|\nabla v|^{2}+m^{2}v^{2}) \,dxdy<\infty.
$$
It is clear that $H^{1}_{m}(A\times \mathbb{R}_{+},y^{1-2s})$ is a Hilbert space with the inner product
$$
\iint_{A\times \mathbb{R}_{+}} y^{1-2s} (\nabla v \nabla z+m^{2}v z) \,dxdy.
$$
When $m=1$, we set  $H^{1}(A\times \mathbb{R}_{+},y^{1-2s})=H^{1}_{1}(A\times \mathbb{R}_{+},y^{1-2s})$.

We denote by $\mathcal{C}^{\infty}_{T}(\mathbb{R}^{N})$ the space of functions
$u\in \mathcal{C}^{\infty}(\mathbb{R}^{N})$ such that $u$ is $T$-periodic in each variable, that is
$$
u(x+e_{i}T)=u(x) \mbox{ for all } x\in \mathbb{R}^{N}, i=1, \dots, N.
$$
Let $u\in  \mathcal{C}^{\infty}_{T}(\mathbb{R}^{N})$. Then we know that
$$
u(x)=\sum_{k\in \mathbb{Z}^{N}} c_{k} \frac{e^{\imath\omega k\cdot x}}{\sqrt{T^{N}}} \quad \mbox{ for all } x\in \mathbb{R}^{N},
$$
where
$$
\omega=\frac{2\pi}{T} \quad \mbox{ and } \quad c_{k}=\frac{1}{\sqrt{T^{N}}} \int_{(0,T)^{N}} u(x)e^{-\imath k\omega \cdot x}dx \quad (k\in \mathbb{Z}^{N})
$$
are the Fourier coefficients of $u$. We define the fractional Sobolev space $\mathbb{H}^{s}_{m,T}$ as the closure of
$\mathcal{C}^{\infty}_{T}(\mathbb{R}^{N})$ under the norm
\begin{equation*}\label{h12norm}
|u|^{2}_{\mathbb{H}^{s}_{m,T}}:= \sum_{k\in \mathbb{Z}^{N}} (\omega^{2}|k|^{2}+m^{2})^{s} \, |c_{k}|^{2}.
\end{equation*}
When $m=1$, we set $\mathbb{H}^{s}_{T}=\mathbb{H}^{s}_{1,T}$ and $|\cdot |_{\mathbb{H}^{s}_{T}}=|\cdot|_{\mathbb{H}^{s}_{1,T}}$.
Now we introduce the functional space $\mathbb{X}^{s}_{m,T}$ defined as the completion of
\begin{align*}
\mathcal{C}_{T}^{\infty}(\overline{\mathbb{R}^{N+1}_{+}})=\Bigl\{ & v\in \mathcal{C}^{\infty}(\overline{\mathbb{R}^{N+1}_{+}}): v(x+e_{i}T,y)=v(x,y) \\
&\mbox{ for every } (x,y)\in \overline{\mathbb{R}_{+}^{N+1}}, i=1, \dots, N \Bigr\}
\end{align*}
under the $H^{1}_{m}(\mathcal{S}_{T},y^{1-2s})$ norm
\begin{equation*}
||v||^{2}_{\mathbb{X}^{s}_{m,T}}:=\iint_{\mathcal{S}_{T}} y^{1-2s} (|\nabla v|^{2}+m^{2}v^{2}) \, dxdy.
\end{equation*}
If $m=1$, we set $\mathbb{X}^{s}_{T}=\mathbb{X}^{s}_{1,T}$ and $||\cdot ||_{\mathbb{X}^{s}_{T}}=||\cdot||_{\mathbb{X}^{s}_{1,T}}$.

We begin proving that it is possible to define a trace operator from $\mathbb{X}^{s}_{m,T}$ to the fractional space $\mathbb{H}^{s}_{m,T}$:

\begin{thm}\label{tracethm}
There exists a bounded linear operator $\textup{Tr} : \mathbb{X}^{s}_{m,T} \rightarrow \mathbb{H}^{s}_{m,T}$  such that :
\begin{itemize}
\item[(i)] $\textup{Tr}(v)=v|_{\partial^{0} \mathcal{S}_{T}}$ for all $v\in \mathcal{C}_{T}^{\infty}(\overline{\mathbb{R}^{N+1}_{+}}) \cap \mathbb{X}^{s}_{m,T}$;
\item[(ii)] There exists $C=C(s)>0$ such that
$$C |\textup{Tr}(v)|_{\mathbb{H}^{s}_{m,T}}\leq ||v||_{\mathbb{X}^{s}_{m,T}} \mbox{ for every } v\in \mathbb{X}^{s}_{m,T};
$$
\item[(iii)] $\textup{Tr}$ is surjective.
\end{itemize}
\end{thm}
\begin{proof}
Let $v\in C^{\infty}_{T}(\overline{\mathbb{R}^{N+1}_{+}})$ such that $\|v\|_{\mathbb{X}^{s}_{m,T}}<\infty$. Then $v$ can be expressed by
$$
v(x,y)= \sum_{k\in \mathbb{Z}^{N}} c_{k}(y) \frac{e^{\imath \omega k\cdot x}}{{\sqrt{T^{N}}}}
$$
where $c_{k}(y)= \int_{(0,T)^{N}} v(x,y) \frac{e^{-\imath \omega k\cdot x}}{{\sqrt{T^{N}}}} \, dx$ and $c_{k}\in H^{1}_{m}(\mathbb{R}_{+}, y^{1-2s})$.

We notice that, by using Parseval's identity, we have
\begin{equation}\label{A0}
\|v\|^{2}_{\mathbb{X}^{s}_{m,T}} = \sum_{k\in \mathbb{Z}^{N}} \int_{0}^{+\infty} y^{1-2s} \, [ (\omega^{2} |k|^{2}+m^{2})  |c_{k}(y)|^{2} +  |c'_{k}(y)|^{2}] \, dy.
\end{equation}

Then, we are going to prove that there exists a positive constant $C_{s}$ depending only on $s$, such that
$$
C_{s} |\textup{Tr}(v)|_{\mathbb{H}^{s}_{m,T}}^{2} \leq \|v\|_{\mathbb{X}^{s}_{m,T}}^{2} \, \mbox{ for any } v\in C^{\infty}_{T}(\overline{\mathbb{R}^{N+1}_{+}}) : \|v\|_{\mathbb{X}^{s}_{m,T}}<+\infty,
$$
or equivalently
\begin{align}\label{A33}
C_{s}& \sum_{k\in \mathbb{Z}^{N}} (\omega^{2} |k|^{2}+m^{2})^{s} |c_{k}(0)|^{2} \nonumber \\
&\leq  \sum_{k\in \mathbb{Z}^{N}} \int_{0}^{+\infty} y^{1-2s} \, [ (\omega^{2} |k|^{2}+m^{2})  |c_{k}(y)|^{2} +  |c'_{k}(y)|^{2}] \, dy.
\end{align}
By the Fundamental Theorem of Calculus we can see
\begin{align*}
|c_{k}(0)| \leq |c_{k}(y)| + \Bigl|\int_{0}^{y} c'_{k}(t) \, dt \Bigr|
\end{align*}
and by using $(|a|+|b|)^{2}\leq 2(|a|^{2}+|b|^{2})$ we have
\begin{align}\label{A11}
|c_{k}(0)|^{2} \leq 2|c_{k}(y)|^{2} + 2 \Bigl|\int_{0}^{y} c'_{k}(t) \, dt \Bigr|^{2}
\end{align}
for any $k\in \mathbb{Z}^{N}$.
Now, we observe that by using H\"older inequality we obtain
\begin{align}\label{A22}
\int_{0}^{y} |c'_{k}(t)|\, dt &\leq \Bigl(\int_{0}^{y} t^{1-2s} |c'_{k}(t)|^{2} dt\Bigr)^{\frac{1}{2}} \Bigl(\int_{0}^{y} t^{2s-1} dt\Bigr)^{\frac{1}{2}} \nonumber \\
&= \Bigl(\int_{0}^{y} t^{1-2s} |c'_{k}(t)|^{2} dt\Bigr)^{\frac{1}{2}} \Bigl(\frac{y^{2s}}{2s}\Bigr)^{\frac{1}{2}}.
\end{align}
Then, putting together (\ref{A11}) and (\ref{A22}) we obtain
\begin{align*}
|c_{k}(0)|^{2} \leq 2|c_{k}(y)|^{2} + \frac{y^{2s}}{s} \Bigl(\int_{0}^{+\infty} t^{1-2s} |c'_{k}(t)|^{2} dt\Bigr)
\end{align*}
and multiplying both sides by $y^{1-2s}$ we get
\begin{align}\label{A3}
y^{1-2s} |c_{k}(0)|^{2} \leq 2 y^{1-2s} |c_{k}(y)|^{2} + \frac{y}{s} \Bigl(\int_{0}^{+\infty} t^{1-2s} |c'_{k}(t)|^{2} dt\Bigr).
\end{align}
Let $a_{k}= (\omega^{2} |k|^{2}+m^{2})^{-\frac{1}{2}}$. Integrating (\ref{A3}) over $y\in (0, a_{k})$ we deduce
\begin{align}\label{A4}
\frac{a_{k}^{2-2s}}{2-2s} |c_{k}(0)|^{2} &\leq 2 \int_{0}^{a_{k}} y^{1-2s} |c_{k}(y)|^{2} dy + \Bigl(\int_{0}^{a_{k}} \frac{y}{s} \, dy \Bigr) \Bigl(\int_{0}^{+\infty} t^{1-2s} |c'_{k}(t)|^{2} dt\Bigr) \nonumber \\
&\leq 2 \int_{0}^{+\infty} y^{1-2s} |c_{k}(y)|^{2} dy + \frac{a_{k}^{2}}{2s} \Bigl(\int_{0}^{+\infty} t^{1-2s} |c'_{k}(t)|^{2} dt\Bigr) \nonumber \\
&= 2 \int_{0}^{+\infty} t^{1-2s} |c_{k}(t)|^{2} dt + \frac{a_{k}^{2}}{2s} \Bigl(\int_{0}^{+\infty} t^{1-2s} |c'_{k}(t)|^{2} dt\Bigr).
\end{align}
Multiplying both sides of (\ref{A4}) by $a_{k}^{-2}= (\omega^{2} |k|^{2}+m^{2})$ we have
\begin{align*}
\frac{(\omega^{2} |k|^{2}+m^{2})^{s}}{2-2s} |c_{k}(0)|^{2} &\leq 2 (\omega^{2} |k|^{2}+m^{2}) \int_{0}^{+\infty} t^{1-2s} |c_{k}(t)|^{2} dt \\
&+ \frac{1}{2s} \Bigl(\int_{0}^{+\infty} t^{1-2s} |c'_{k}(t)|^{2} dt\Bigr)
\end{align*}
for any $k\in \mathbb{Z}^{N}$.

Summing over $\mathbb{Z}^{N}$ we deduce
\begin{align}\label{A5}
\frac{1}{2-2s}  &\sum_{k\in \mathbb{Z}^{N}} (\omega^{2} |k|^{2}+m^{2})^{s} |c_{k}(0)|^{2} \leq \nonumber \\
&\leq  \sum_{k\in \mathbb{Z}^{N}} \Bigl[2(\omega^{2} |k|^{2}+m^{2}) \int_{0}^{+\infty} t^{1-2s} |c_{k}(t)|^{2} dt
+ \frac{1}{2s} \Bigl(\int_{0}^{+\infty} t^{1-2s} |c'_{k}(t)|^{2} dt\Bigr)\Bigr] \nonumber \\
&\leq \max \Bigl\{2, \frac{1}{2s}\Bigr \} \sum_{k\in \mathbb{Z}^{N}} \int_{0}^{+\infty} t^{1-2s} \, [ (\omega^{2} |k|^{2}+m^{2})  |c_{k}(t)|^{2} +  |c'_{k}(t)|^{2}] \, dt.
\end{align}
Taking into account (\ref{A0}) and (\ref{A5}) we have $(\ref{A33})$.

Therefore there exists a trace operator $\textup{Tr} : \mathbb{X}^{s}_{m,T} \rightarrow \mathbb{H}^{s}_{m,T}$.
Now we prove that $\textup{Tr}$ is surjective. Let $u= \sum_{k\in \mathbb{Z}^{N}} c_{k} \frac{e^{\imath \omega k\cdot x}}{{\sqrt{T^{N}}}} \in \mathbb{H}^{s}_{m,T}$.

We define
\begin{align}\label{extu}
v(x,y)=\sum_{k\in \mathbb{Z}^{N}} c_{k} \theta_{k}(y) \frac{e^{\imath \omega k\cdot x}}{{\sqrt{T^{N}}}}
\end{align}
where $\theta_{k}(y)=  \theta(\sqrt{\omega^{2} |k|^{2}+ m^{2}} y)$ and $\theta(y)\in H^{1}(\mathbb{R}_{+},y^{1-2s})$ solves the following ODE
\begin{equation*} \label{ccv}
\left\{
\begin{array}{cc}
&\theta^{''}+\frac{1-2s}{y}\theta^{'}-\theta=0 \mbox{ in } \mathbb{R}_{+}  \\
&\theta(0)=1 \mbox{ and } \theta(\infty)=0
\end{array}.
\right.
\end{equation*}
It is known (see \cite{Erd}) that $\theta(y)=\frac{2}{\Gamma(s)} (\frac{y}{2})^{s} K_{s}(y)$ where $K_{s}$ is the Bessel function of second kind with order $s$, and
being $K'_{s}=\frac{s}{y}K_{s}-K_{s-1}$, we can see that
$$
\kappa_{s}:= \int_{0}^{\infty} y^{1-2s} (|\theta'(y)|^{2}+|\theta(y)|^{2}) dy=-\lim_{y \rightarrow 0} y^{1-2s}\theta'(y)=2^{1-2s} \frac{\Gamma(1-s)}{\Gamma(s)}.
$$
Then it is clear that $v$ is smooth for $y>0$, $v$ is $T$-periodic in $x$ and satisfies
$$
-\dive(y^{1-2s} \nabla v)+m^{2}y^{1-2s}v =0 \mbox{ in } \mathcal{S}_{T} .
$$
Now, we show that $\textup{Tr}(v)=u$. From standard properties of $K_{s}$, we know $\theta(y) \rightarrow 1$ as $y \rightarrow 0$ and
$0<\theta(y)\leq A_{s}$ for any $y \geq 0$.
Then, being $u\in \mathbb{H}^{s}_{m,T}$, we have
$$
|v(\cdot,y)-u|_{\mathbb{H}^{s}_{m,T}}^{2}=\sum_{k\in \mathbb{Z}^{N}}(\omega^{2}|k|^{2}+m^{2})^{s} |c_{k}|^{2} |\theta_{k}(y)-1|^{2} \rightarrow 0 \mbox{ as } y\rightarrow 0.
$$
Finally we prove that $v\in \mathbb{X}^{s}_{m,T}$. By Parseval's identity we get
\begin{align}
||v||^{2}_{\mathbb{X}^{s}_{m,T}}&=\iint_{\mathcal{S}_{T}} y^{1-2s} (|\nabla v|^{2} + m^{2} v^{2} ) \, dxdy \nonumber \\
&=\sum_{k\in \mathbb{Z}^{N}} |c_{k}|^{2} \int_{0}^{\infty} y^{1-2s} (|\theta'_{k}(y)|^{2} +|\theta_{k}(y)|^{2} ) dy  \nonumber \\
&=\sum_{k\in \mathbb{Z}^{N}} |c_{k}|^{2} \int_{0}^{\infty} y^{1-2s} (\omega^{2} |k|^{2}+m^{2}) (|\theta'(\sqrt{\omega^{2} |k|^{2}+m^{2}}y)|^{2} +|\theta(\sqrt{\omega^{2} |k|^{2}+m^{2}}y)|^{2} ) dy  \nonumber \\
&=\sum_{k\in \mathbb{Z}^{N}} |c_{k}|^{2} \frac{\sqrt{\omega^{2} |k|^{2}+m^{2}}}{(\omega^{2} |k|^{2}+m^{2})^{\frac{1-2s}{2}}} \int_{0}^{\infty} y^{1-2s} (|\theta'(y)|^{2} +|\theta(y)|^{2} )dy  \nonumber \\
&=\kappa_{s} \sum_{k\in \mathbb{Z}^{N}} (\omega^{2} |k|^{2}+m^{2})^{s} |c_{k}|^{2}  \nonumber \\
&=\kappa_{s} |u|^{2}_{\mathbb{H}^{s}_{m,T}}. \label{v=u}
\end{align}

\end{proof}

Now we prove the following embedding
\begin{thm}\label{thm2}
Let $N> 2s$. Then  $\textup{Tr}(\mathbb{X}^{s}_{m,T})$ is continuously embedded in $L^{q}(0,T)^{N}$ for any  $1\leq q \leq 2^{\sharp}_{s}$.  Moreover,  $\textup{Tr}(\mathbb{X}^{s}_{m,T})$ is compactly embedded in $L^{q}(0,T)^{N}$  for any  $1\leq q < 2^{\sharp}_{s}$.
\end{thm}
\begin{proof}
By Theorem \ref{tracethm} we know that there exists a continuous embedding from $\mathbb{X}^{s}_{m,T}$ to $\mathbb{H}^{s}_{m,T}$.
Now, we prove that $ \mathbb{H}^{s}_{m,T}$ is continuously embedded in $L^{q}(0,T)^{N}$ for any $q \leq 2^{\sharp}_{s}$ and compactly in $L^{q}(0,T)^{N}$ for any $q< 2^{\sharp}_{s}$.

By using Proposition $2.1.$ in \cite{benyi}, we know that there exists a constant $C_{2^{\sharp}_{s}}>0$ such that
\begin{align}\label{benyi}
|u|_{L^{2^{\sharp}_{s}}(0,T)^{N}} \leq C_{2^{\sharp}_{s}} \Bigl( \sum_{|k|\geq 1} \omega^{2s}|k|^{2s} |c_{k}|^{2} \Bigr)^{\frac{1}{2}}
\end{align}
for any $u\in \mathcal{C}^{\infty}_{T}(\mathbb{R}^{N})$ such that $\frac{1}{T^{N}}\int_{(0,T)^{N}} u(x) \, dx =0$.
As a consequence, fixed $2 \leq q\leq 2^{\sharp}_{s}$, we have
\begin{equation}\label{6.22}
|u|_{L^{q}(0,T)^{N}}\leq C \Bigl( \sum_{k\in \mathbb{Z}^{N}} |c_{k}|^{2} (\omega^{2}|k|^{2}+m^{2})^{s}   \Bigr)^{\frac{1}{2}}
\end{equation}
for any $u\in \mathbb{H}^{s}_{m,T}$, that is $\mathbb{H}^{s}_{m,T}$ is continuously embedded in $L^{q}(0,T)^{N}$ for any $2 \leq q\leq 2^{\sharp}_{s}$.
Now, we proceed as the proof of Theorem $4$ in \cite{A2} to prove that $\mathbb{H}^{s}_{m,T} \Subset L^{q}(0,T)^{N}$ for any $2 \leq q<2^{\sharp}_{s}$.
Fix $q\in [2,2^{\sharp}_{s})$. Then, by using  (\ref{6.22}) and the interpolation inequality we obtain
\begin{equation}\label{6.23}
|u|_{L^{q}(0,T)^{N}}\leq C|u|^{\theta}_{L^{2}(0,T)^{N}} \Bigl( \sum_{k\in \mathbb{Z}^{N}} |c_{k}|^{2} (\omega^{2}|k|^{2}+m^{2})^{s}   \Bigr)^{1-\theta},
\end{equation}
for some real positive number $\theta\in (0,1)$.

Now, taking into account (\ref{6.23}), it is enough to prove that $\mathbb{H}^{s}_{m,T}\Subset L^{2}(0,T)^{N}$ to infer that $\mathbb{H}^{s}_{m,T}$ is compactly embedded in $L^{q}(0,T)^{N}$ for every $q\in [2,2^{\sharp}_{s})$.

Let us assume that $u^{j}\rightharpoonup 0$ in $\mathbb{H}^{s}_{m,T}$. Then
\begin{equation}\label{6.24}
\lim_{j\rightarrow \infty} |c^{j}_{k}|^{2}(\omega^{2}|k|^{2}+m^{2})^{s}=0 \quad \forall k\in \mathbb{Z}^{N}
\end{equation}
and
\begin{equation}\label{6.25}
 \sum_{k\in \mathbb{Z}^{N}}  |c^{j}_{k}|^{2} (\omega^{2}|k|^{2}+m^{2})^{s}\leq C \quad \forall j\in \mathbb{N}.
\end{equation}
Fix $\varepsilon>0$. Then there exists $\nu_{\varepsilon}>0$ such that $(\omega^{2}|k|^{2}+m^{2})^{-s}<\varepsilon$ for $|k|>\nu_{\varepsilon}$.
By (\ref{6.25}) we have
\begin{align*}
\sum_{k\in \mathbb{Z}^{N}} |c_{k}^{j}|^{2}&=\sum_{|k|\leq \nu_{\varepsilon}} |c_{k}^{j}|^{2}+\sum_{|k|>\nu_{\varepsilon}} |c_{k}^{j}|^{2} \\
&= \sum_{|k|\leq \nu} |c_{k}^{j}|^{2}+\sum_{|k|>\nu_{\varepsilon}} |c_{k}^{j}|^{2}(\omega^{2}|k|^{2}+m^{2})^{s} (\omega^{2}|k|^{2}+m^{2})^{-s} \\
&\leq \sum_{|k|\leq \nu_{\varepsilon}} |c_{k}^{j}|^{2}+C\varepsilon.
\end{align*}
Using (\ref{6.24}) we deduce that $\sum_{|k|\leq \nu_{\varepsilon}} |c_{k}^{j}|^{2}<\varepsilon$ for $j$ large. Thus $u^{j}\rightarrow 0$ in $L^{2}(0,T)^{N}$.

\end{proof}

Now, we conclude this section giving some elementary results about the nonlinearity $f(x,t)$. More precisely, by using the assumptions $(f2), (f3)$ and $(f4)$, we can deduce some bounds from above and below for the nonlinear term $f(x,t)$ and its primitive $F(x,t)$. This part is quite standard and the proofs of the following Lemmas can be found, for instance, in \cite{AR} and \cite{Rab}.
\begin{lem}\label{lfF}
Let $f:[0,T]^{N}\times \mathbb{R} \rightarrow \mathbb{R}$ satisfying conditions $(f1)$-$(f3)$. Then, for any $\varepsilon>0$ there
exists $C_{\varepsilon}>0$ such that
\begin{equation*}\label{f}
|f(x,t)|\leq  2\varepsilon|t|+(p+1)C_{\varepsilon}|t|^{p} \quad \forall t\in{\mathbb{R}} \, \forall x\in [0,T]^{N}
\end{equation*}
and
\begin{equation}\label{F}
|F(x,t)|\leq  \varepsilon|t|^{2}+C_{\varepsilon}|t|^{p+1} \quad \forall t\in {\mathbb{R}} \, \forall x\in [0,T]^{N}.
\end{equation}
\end{lem}
\begin{lem}\label{F**}
Assume that $f:[0,T]^{N}\times \mathbb{R} \rightarrow \mathbb{R}$ satisfies conditions $(f1)$-$(f4)$. Then, there exist two constants $a_{3}>0$ and $a_{4}>0$  such that
\begin{equation*}\label{F*}
F(x,t)\geq a_{3}|t|^{\mu}-a_{4} \quad \forall t\in{\mathbb{R}} \, \forall x\in [0, T]^{N}.
\end{equation*}
\end{lem}

\section{Extension problem}

In this section we show that to study $(\ref{P})$ it is equivalent to investigate the solutions of a problem in a half-cylinder with a Neumann nonlinear boundary condition.
We start proving that

\begin{thm}
Let $u\in \mathbb{H}^{s}_{m,T}$. Then there exists a unique $v\in \mathbb{X}^{s}_{m,T}$ such that
\begin{equation}\label{extPu}
\left\{
\begin{array}{ll}
-\dive(y^{1-2s} \nabla v)+m^{2}y^{1-2s}v =0 &\mbox{ in }\mathcal{S}_{T}  \\
v_{| {\{x_{i}=0\}}}= v_{| {\{x_{i}=T\}}} & \mbox{ on } \partial_{L}\mathcal{S}_{T} \\
v(\cdot,0)=u  &\mbox{ on } \partial^{0}\mathcal{S}_{T}
\end{array}
\right.
\end{equation}
and
\begin{align}\label{conormal}
-\lim_{y \rightarrow 0} y^{1-2s}\frac{\partial v}{\partial y}(x,y)=\kappa_{s} (-\Delta_{x}+m^{2})^{s}u(x) \mbox{ in } \mathbb{H}^{-s}_{m,T},
\end{align}
where
$$
\mathbb{H}^{-s}_{m,T}=\Bigl \{u=\sum_{k\in \mathbb{Z}^{N}} c_{k} \frac{e^{\imath \omega k\cdot x}}{{\sqrt{T^{N}}}}: \frac{|c_{k}|^{2}}{(\omega^{2}|k|^{2}+m^{2})^{s}}< \infty \Bigr\}
$$
is the dual of $\mathbb{H}^{s}_{m,T}$.
\end{thm}
\begin{proof}
Let $u=\sum_{k\in \mathbb{Z}^{N}} c_{k} \frac{e^{\imath \omega k\cdot x}}{{\sqrt{T^{N}}}}\in \mathcal{C}^{\infty}_{T}(\mathbb{R}^{N})$.
Let us consider the following problem
\begin{equation}\label{minpb}
\min\{ ||v||^{2}_{\mathbb{X}^{s}_{m,T}}: v\in \mathbb{X}^{s}_{m,T}, \textup{Tr}(v)=u \}.
\end{equation}
By using the Theorem \ref{thm2}, we can find a minimizer to $(\ref{minpb})$.
Since $||\cdot||^{2}_{\mathbb{X}^{s}_{m,T}}$ is strictly convex, such minimizer is unique and we denote it by $v$.
As a consequence, for any $\phi \in \mathbb{X}^{s}_{m,T}$ such that $\textup{Tr}(\phi)=0$
\begin{equation}\label{WE}
\iint_{\mathcal{S}_{T}} y^{1-2s} (\nabla v \nabla \phi + m^{2} v \phi  ) \, dxdy=0,
\end{equation}
that is $v$ is a weak solution to $(\ref{extPu})$.
Since the function defined in (\ref{extu}) is a solution to (\ref{extPu}), by the uniqueness of minimizer, we deduce that $v$ is given by
$$
v(x,y)=\sum_{k\in \mathbb{Z}^{N}} c_{k} \theta_{k}(y) \frac{e^{\imath \omega k\cdot x}}{{\sqrt{T^{N}}}}.
$$
where $\theta_{k}(y)=\theta(\sqrt{\omega^{2}|k|^{2}+ m^{2}} y)$. In particular, by (\ref{v=u}), we have
$$
||v||_{\mathbb{X}^{s}_{m,T}}=\sqrt{\kappa_{s}} |u|_{\mathbb{H}^{s}_{m,T}}.
$$

Then
\begin{align*}
&\Bigl| -y^{1-2s} \frac{\partial v}{\partial y}(\cdot,y)-\kappa_{s} (-\Delta+m^{2})^{s}u \Bigr|^{2}_{\mathbb{H}^{-s}_{m,T}} \\
&=\sum_{k\in \mathbb{Z}^{N}} \frac{1}{(\omega^{2}|k|^{2}+m^{2})^{s}} |c_{k}|^{2} \Bigl|\sqrt{\omega^{2}|k|^{2}+m^{2}} \theta_{k}'(y)y^{1-2s}+\kappa_{s}(\omega^{2}|k|^{2}+m^{2})^{s} \Bigr|^{2} \\
&=\sum_{k\in \mathbb{Z}^{N}} (\omega^{2}|k|^{2}+m^{2})^{s}  |c_{k}|^{2} \Bigl|[(\omega^{2}|k|^{2}+m^{2})y]^{1-2s} \theta_{k}'(y)+\kappa_{s} \Bigr|^{2}
\end{align*}
and by using $u\in \mathbb{H}^{s}_{m,T}$, $-y^{1-2s} \theta'(y) \rightarrow \kappa_{s}$ as $y\rightarrow 0$ and $0<-\kappa_{s}y^{1-2s} \theta'(y)\leq B_{s}$ for any $y \geq 0$ (see \cite{Erd}), we deduce (\ref{conormal}).

\end{proof}

Therefore, for any given $u\in \mathbb{H}^{s}_{m,T}$ we can find a unique function $v=\textup{Ext}(u)\in \mathbb{X}^{s}_{m,T}$, which will be called the extension of $u$, such that
\begin{compactenum}[(i)]
\item $v$ is smooth for $y>0$, $T$-periodic in $x$ and $v$ solves (\ref{extPu});
\item $||v||_{\mathbb{X}^{s}_{m,T}}\leq ||z||_{\mathbb{X}^{s}_{m,T}}$ for any $z\in \mathbb{X}^{s}_{m,T}$ such that $\textup{Tr}(z)=u$;
\item $||v||_{\mathbb{X}^{s}_{m,T}}=\sqrt{\kappa_{s}} |u|_{\mathbb{H}^{s}_{m,T}}$;
\item We have
$$
\lim_{y\rightarrow 0}-y^{1-2s} \frac{\partial v}{\partial y}(x,y)= \kappa_{s} (-\Delta+m^{2})^{s}u(x) \mbox{ in } \mathbb{H}^{-s}_{m,T}.
$$
\end{compactenum}

Now, modifying the proof of Lemma $2.2$ in \cite{CapDavDupSir} we deduce
\begin{thm}
Let $g\in \mathbb{H}^{-s}_{m,T}$. Then, there is a unique solution to the problem:
$$
\mbox{ find } u\in \mathbb{H}^{s}_{m,T} \quad \mbox{ such that } \quad (-\Delta+m^{2})^{s} u=g.
$$
Moreover $u$ is the trace of $v\in \mathbb{X}^{s}_{m,T}$, where $v$ is the unique solution to (\ref{R}), that is
for every $\phi \in \mathbb{X}^{s}_{m,T}$ it holds
$$
\iint_{\mathcal{S}_{T}} y^{1-2s} (\nabla v \nabla \phi + m^{2} v \phi  ) \, dxdy=\kappa_{s}\langle g, \textup{Tr}(\phi)\rangle_{\mathbb{H}^{-s}_{m,T}, \mathbb{H}^{s}_{m,T}}.
$$
\end{thm}

Taking into account the previous results we can reformulate the nonlocal problem (\ref{P}) in a local way as explained below.

Let $g \in \mathbb{H}^{-s}_{m,T}$ and consider the following two problems:
\begin{equation}\label{P*}
\left\{
\begin{array}{ll}
(-\Delta_{x}+m^{2})^{s}u=g &\mbox{ in } (0,T)^{N}  \\
u(x+Te_{i})=u(x) & \mbox{ for } x\in \mathbb{R}^{N}
\end{array}
\right.
\end{equation}
and
\begin{equation}\label{P**}
\left\{
\begin{array}{ll}
-\dive(y^{1-2s} \nabla v)+m^{2}y^{1-2s}v =0 &\mbox{ in }\mathcal{S}_{T}  \\
v_{| {\{x_{i}=0\}}}= v_{| {\{x_{i}=T\}}} & \mbox{ on } \partial_{L}\mathcal{S}_{T} \\
\frac{\partial v}{\partial \nu^{1-2s}}=g(x)  &\mbox{ on } \partial^{0}\mathcal{S}_{T}
\end{array}.
\right.
\end{equation}

\begin{defn}
We say that $u\in \mathbb{H}^{s}_{m,T}$ is a weak solution to (\ref{P*}) if $u=\textup{Tr}(v)$
and $v$ is a weak solution to (\ref{P**}).
\end{defn}

\begin{remark}
Later, with abuse of notation, we will denote by $v(\cdot,0)$ the trace $\textup{Tr}(v)$ of a function $v\in \mathbb{X}^{s}_{m,T}$.
\end{remark}

We conclude this section giving the proof of the following sharp trace inequality:
\begin{thm}\label{thm6}
For any $v\in \mathbb{X}^{s}_{m,T}$ we have
\begin{equation}\label{eqYY}
 \kappa_{s} |\textup{Tr}(v)|_{\mathbb{H}^{s}_{m,T}}^{2}\leq ||v||_{\mathbb{X}^{s}_{m,T}}^{2}
\end{equation}
and the equality is attained if and only if $v=\textup{Ext}(\textup{Tr}(v))$.
In particular
\begin{equation}\label{eqY}
||v||_{\mathbb{X}^{s}_{m,T}}^{2} - \kappa_{s} m^{2s} |\textup{Tr}(v)|_{L^{2}(0,T)^{N}}^{2} =0 \Leftrightarrow v(x,y)=C\, \theta(my) \mbox{ for some } C\in \mathbb{R}.
\end{equation}
\end{thm}
\begin{proof}
By properties $(ii)$ and $(iii)$ we can see that for any $v\in \mathbb{X}^{s}_{m,T}$
\begin{equation*}
 \kappa_{s} |\textup{Tr}(v)|_{\mathbb{H}^{s}_{m,T}}^{2}=||\textup{Ext}\textup{Tr}(v)||_{\mathbb{X}^{s}_{m,T}}^{2}\leq ||v||_{\mathbb{X}^{s}_{m,T}}^{2},
\end{equation*}
and the equality holds if and only if $v=\textup{Ext}\textup{Tr}(v)$.

Now, we prove (\ref{eqY}).
We denote by $c_{k}$ the Fourier coefficients of $\textup{Tr}(v)$.

If $v(x,y)= C \theta_{m}(y) : = C \theta(my)$ for some $C\in \mathbb{R}$, being $\theta(0)=1$, we have
\begin{align*}
||v||_{\mathbb{X}^{s}_{m,T}}^{2}&= C^{2} T^{N}\int_{0}^{+\infty} y^{1-2s} (|\theta'_{m}(y)|^{2}+m^{2} |\theta_{m}(y)|^{2})\, dy\\
&= C^{2} T^{N}m^{2s} \int_{0}^{+\infty} y^{1-2s} (|\theta'(y)|^{2}+m^{2} |\theta(y)|^{2})\, dy\\
&= C^{2} T^{N}m^{2s} \kappa_{s}\\
&= m^{2s} \kappa_{s} |\textup{Tr}(v)|_{L^{2}(0,T)^{N}}^{2}.
\end{align*}
Now, we assume that
$$
||v||_{\mathbb{X}^{s}_{m,T}}^{2} - \kappa_{s} m^{2s} |\textup{Tr}(v)|_{L^{2}(0,T)^{N}}^{2} =0.
$$
By using $(ii)$ and $(iii)$ we deduce
\begin{align}
||\textup{Ext}\textup{Tr}(v)||_{\mathbb{X}^{s}_{m,T}}^{2}&\leq
||v||_{\mathbb{X}^{s}_{m,T}}^{2}= \kappa_{s} m^{2s} |\textup{Tr}(v)|_{L^{2}(0,T)^{N}}^{2} \nonumber \\
&\leq \kappa_{s} |\textup{Tr}(v)|_{\mathbb{H}^{s}_{m,T}}^{2}=||\textup{Ext}\textup{Tr}(v)||_{\mathbb{X}^{s}_{m,T}}^{2}
\end{align}
that is
$$
||v||_{\mathbb{X}^{s}_{m,T}}^{2}=||\textup{Ext}\textup{Tr}(v)||_{\mathbb{X}^{s}_{m,T}}^{2}=\kappa_{s} m^{2s} |\textup{Tr}(v)|_{L^{2}(0,T)^{N}}^{2}.
$$
Let us note that $||v||_{\mathbb{X}^{s}_{m,T}}^{2}=||\textup{Ext}\textup{Tr}(v)||_{\mathbb{X}^{s}_{m,T}}^{2}$ implies $v=\textup{Ext}(\textup{Tr}(v))$.

In particular, from
$$
\kappa_{s}m^{2s} |\textup{Tr}(v)|_{L^{2}(0,T)^{N}}^{2}=||\textup{Ext}\textup{Tr}(v)||_{\mathbb{X}^{s}_{m,T}}^{2}=\kappa_{s} |\textup{Tr}(v)|_{\mathbb{H}^{s}_{m,T}}^{2}
$$
we obtain that $c_{k}=0$  for any $k\neq 0$, so we get
$$
v=\textup{Ext}(\textup{Tr}(v))=\sum_{k\in \mathbb{Z}^{N}} c_{k}\theta(\sqrt{\omega^{2} |k|^{2}+ m^{2}} y) e^{\imath k \cdot x}=c_{0} \theta(my).
$$

\end{proof}

\section{Periodic solutions in the cylinder $\mathcal{S}_{T}$}

In this section we prove the existence of a solution to $(\ref{P})$. As shown in previous section, we know that the study of $(\ref{P})$ is equivalent to investigate the existence of weak solutions to
\begin{equation}
\left\{
\begin{array}{ll}
-\dive(y^{1-2s} \nabla v)+m^{2}y^{1-2s}v =0 &\mbox{ in }\mathcal{S}_{T}:=(0,T)^{N} \times (0,\infty)  \\
v_{| {\{x_{i}=0\}}}= v_{| {\{x_{i}=T\}}} & \mbox{ on } \partial_{L}\mathcal{S}_{T}:=\partial (0,T)^{N} \times [0,\infty) \\
\frac{\partial v}{\partial \nu^{1-2s}}=\kappa_{s} [m^{2s}v+f(x,v)]   &\mbox{ on }\partial^{0}\mathcal{S}_{T}:=(0,T)^{N} \times \{0\}
\end{array}.
\right.
\end{equation}
For simplicity, we will assume that $\kappa_{s}=1$.

Then, we will look for the critical points of
$$
\mathcal{J}_{m}(v)=\frac{1}{2}||v||^{2}_{\mathbb{X}^{s}_{m,T}} -\frac{ m^{2s}}{2}|v(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}- \int_{\partial^{0}\mathcal{S}_{T}} F(x,v) dx
$$
defined for $v\in\mathbb{X}^{s}_{m,T}$.

More precisely, we will prove that $\mathcal{J}_{m}$ satisfies the assumptions of the Linking Theorem \cite{Rab}:
\begin{thm}\label{Linking Thm}
Let $(X, ||\cdot||)$ be a real Banach space with $X=Y\bigoplus Z$, where $Y$ is finite dimensional. Let $J\in \mathcal{C}^{1}(X,\mathbb{R})$ be a functional satisfying the following conditions:
\begin{itemize}
\item $J$ satisfies the Palais-Smale condition,
\item There exist $\eta, \rho>0$ such that
$$
\inf \{J(v): v\in Z \mbox{ and } ||v||=\eta\} \geq \rho,
$$
\item There exist $z\in \partial B_{1} \cap Z$, $R>\eta$ and $R'>0$ such that
$$
 J\leq 0 \mbox{ on } \partial \mathcal{A}
$$
where
$$
\mathcal{A}=\{v=y+r z: y\in Y, ||y||\leq R' \mbox{ and } 0\leq r \leq R\}
$$
and
$$
\partial \mathcal{A}=\{v=y+r z: y\in Y, ||y||= R' \mbox{ or }  r\in \{0, R\} \}.
$$
\end{itemize}
Then $J$ possesses a critical value $c\geq \rho$ which can be characterized as
$$
c:=\inf_{\gamma \in \Gamma} \max_{v\in \mathcal{A}} J(\gamma(v))
$$
where
$$
\Gamma:=\{\gamma \in \mathcal{C}(\mathcal{A}, X): \gamma=Id \mbox{ on } \partial \mathcal{A}\}.
$$
\end{thm}

By using the assumptions on $f$, it is easy to prove that  $\mathcal{J}_{m}$ is well defined on $\mathbb{X}^{s}_{m,T}$ and $\mathcal{J}_{m}\in \mathcal{C}^{1}(\mathbb{X}^{s}_{m,T},\mathbb{R})$. Moreover, by (\ref{eqYY}) we notice that the quadratic part of $\mathcal{J}_{m}$ is nonnegative, that is
\begin{equation}\label{partequad}
||v||^{2}_{\mathbb{X}^{s}_{m,T}} -m^{2s}|v(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}\geq 0.
\end{equation}

Let us note that
$$
\mathbb{X}^{s}_{m,T}=<\theta(my)> \oplus \Bigl\{ v\in \mathbb{X}^{s}_{m,T}: \int_{(0,T)^{N}} v(x,0) dx=0\Bigr \}=:\mathbb{Y}^{s}_{m,T} \oplus \mathbb{Z}^{s}_{m,T}
$$
where $\dim \mathbb{Y}^{s}_{m,T}<\infty$ and $\mathbb{Z}^{s}_{m,T}$ is the orthogonal complement of $\mathbb{Y}^{s}_{m,T}$ with respect to the inner product in $\mathbb{X}^{s}_{m,T}$.
In order to prove that $\mathcal{J}_{m}$ verifies the Linking hypotheses we prove the following Lemmas
\begin{lem}\label{Linking1}
$\mathcal{J}_{m}\leq 0 \mbox{ on } \mathbb{Y}^{s}_{m,T}$.
\end{lem}
\begin{proof}
It follows directly by (\ref{eqY}) and by assumption $(f6)$.

\end{proof}

\begin{lem}
There exist $\rho>0$ and $\eta>0$ such that
$\mathcal{J}_{m}(v)\geq \rho \mbox{ for } v\in \mathbb{Z}^{s}_{m,T}: ||v||_{\mathbb{X}^{s}_{m,T}}=\eta$.
\end{lem}
\begin{proof}
Firstly we show that there exists a constant $C>0$ such that
\begin{align}\label{eqnorm}
||v||_{\mathbb{X}^{s}_{m,T}}^{2}-m^{2s}|v(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}\geq C||v||_{\mathbb{X}^{s}_{m,T}}^{2}
\end{align}
for any $v\in \mathbb{Z}^{s}_{m,T}$.
Assume by contradiction that there exists a sequence $(v_{j})\subset \mathbb{Z}^{s}_{m,T}$ such that
$$
||v_{j}||_{\mathbb{X}^{s}_{m,T}}^{2}-m^{2s}|v_{j}(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}< \frac{1}{j} ||v_{j}||_{\mathbb{X}^{s}_{m,T}}^{2}
$$
Let $z_{j}=v_{j}/||v_{j}||_{\mathbb{X}^{s}_{m,T}}$. Then $||z_{j}||_{\mathbb{X}^{s}_{m,T}}=1$ so we can assume that $z_{j}\rightharpoonup z$ in $\mathbb{X}^{s}_{m,T}$ and $z_{j}(\cdot,0) \rightarrow z(\cdot,0)$ in $L^{2}(0,T)^{N}$ for some $z\in \mathbb{Z}^{s}_{m,T}$ ($\mathbb{Z}^{s}_{m,T}$ is weakly closed).

Hence, for any $j\in \mathbb{N}$
$$
1-m^{2s}|z_{j}(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}<\frac{1}{j}
$$
so  we get $|z_{j}(\cdot,0)|^{2}_{L^{2}(0,T)^{N}}\rightarrow \frac{1}{m^{2s}}$ that is $|z(\cdot,0)|_{L^{2}(0,T)^{N}}=\frac{1}{m^{s}}$.

On the other hand
\begin{align*}
0&\leq ||z||_{\mathbb{X}^{s}_{m,T}}^{2}-m^{2s}|z(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}\\
&\leq \liminf_{j\rightarrow \infty}||z_{j}||_{\mathbb{X}^{s}_{m,T}}^{2}-m^{2s}|z_{j}(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}=0
\end{align*}
implies that $z=c\, \theta(my)$ by (\ref{eqY}). But $z\in \mathbb{Z}^{s}_{m,T}$ so $c=0$ and this is a contradiction because of $|z(\cdot,0)|_{L^{2}(0,T)^{N}}=\frac{1}{m^{s}}>0$.

Taking into account $(\ref{eqnorm})$, $(\ref{F})$ and Theorem $4$ we have
\begin{align*}
\mathcal{J}_{m}(v)&\geq C||v||_{\mathbb{H}^{s}_{m,T}}^{2}-\varepsilon|v(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}-C_{\varepsilon} |v(\cdot,0)|_{L^{p+1}(0,T)^{N}}^{p+1} \\
&\geq \Bigl(C-\frac{\varepsilon}{m}\Bigr)||v||_{\mathbb{X}^{s}_{m,T}}^{2}-C||v||_{\mathbb{X}^{s}_{m,T}}^{p+1}
\end{align*}
for any $v\in \mathbb{Z}^{s}_{m,T}$.
Choosing $\varepsilon \in (0,mC)$, we can find $\rho>0$ and $\eta>0$ such that
$$
\inf \{\mathcal{J}_{m}(v):  v\in \mathbb{Z}^{s}_{m,T} \mbox{ and } ||v||_{\mathbb{X}^{s}_{m,T}}=\eta  \}\geq \rho.
$$

\end{proof}

\begin{lem}\label{lemma5}
There exist $R>\rho$, $R'>0$ and $z\in \mathbb{Z}^{s}_{m,T}$ such that
$$
\max_{\partial \mathbb{A}^{s}_{m,T}} \mathcal{J}_{m}(v) \leq 0 \quad \mbox{ and } \quad \max_{ \mathbb{A}^{s}_{m,T}} \mathcal{J}_{m}(v) <\infty
$$
where
$$
\mathbb{A}^{s}_{m,T}=\{ v=y+r z: ||y||_{\mathbb{X}^{s}_{m,T}} \leq R' \mbox{ and }  r\in [0, R] \}.
$$
\end{lem}
\begin{proof}
By using Lemma \ref{Linking1} we know that $\mathcal{J}_{m}\leq 0$ on $\mathbb{Y}^{s}_{m,T}$.

Let us consider
$$
w=\prod_{i=1}^{N} \sin(\omega x_{i}) \frac{1}{y+1}.
$$
We note that $w\in \mathbb{Z}^{s}_{m,T}$ (since $\int_{0}^{T} \sin(\omega x) dx=0$) and
\begin{align*}
||w||^{2}_{\mathbb{X}^{s}_{m,T}}&=N\Bigl(\prod_{i=1}^{N-1} \int_{0}^{T} \sin^{2}(\omega x) dx\Bigr)
\omega^{2} \Bigl( \int_{0}^{T} \cos^{2}(\omega x) dx\Bigr)\Bigl(\int_{0}^{\infty} y^{1-2s} \frac{dy}{(y+1)^{2}} \Bigr) \\
&+\Bigl(\prod_{i=1}^{N} \int_{0}^{T} \sin^{2}(\omega x) dx \Bigr) \Bigl(\int_{0}^{\infty} y^{1-2s} \frac{dy}{(y+1)^{4}} \Bigr)\\
&+m^{2}\Bigl(\prod_{i=1}^{N} \int_{0}^{T} \sin^{2}(\omega x) dx \Bigr) \Bigl(\int_{0}^{\infty} y^{1-2s} \frac{dy}{(y+1)^{2}} \Bigr).
\end{align*}
So there exist $C_{1}, C_{2}, C_{3}>0$ (independent of $m$) such that
\begin{equation}\label{einstein}
C_{1}\leq ||w||^{2}_{\mathbb{X}^{s}_{m,T}}\leq C_{2}+m^{2}C_{3}.
\end{equation}
We set  $z=\frac{w}{||w||_{\mathbb{X}^{s}_{m,T}}}$.
It is clear that $z\in \mathbb{Z}^{s}_{m,T}$ and $||z||_{\mathbb{X}^{s}_{m,T}}=1$.

By H\"older inequality, we can observe that if $v=y+rz\in \mathbb{Y}^{s}_{m,T}\oplus \mathbb{R}_{+}z$
\begin{align*}
|v(\cdot,0)|_{L^{\mu}(0,T)^{N}}^{\mu} &\geq C|v(\cdot,0)|_{L^{2}(0,T)^{N}}^{\mu} \\
&=C\Bigl(\int_{(0,T)^{N}} (c+rz)^{2} dx \Bigr)^{\frac{\mu}{2}} \\
& \geq  C'(m^{2s}c^{2}T^{N}+r^{2})^{\frac{\mu}{2}}.
\end{align*}

Then, for any $v=y+r z\in \mathbb{Y}^{s}_{m,T}\oplus \mathbb{R}_{+} z$
\begin{align}
\mathcal{J}_{m}(v)&=\frac{1}{2}||z||_{\mathbb{X}^{s}_{m,T}}^{2}-\frac{m^{2s}}{2}|z(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}-\int_{\partial^{0}\mathcal{S}_{T}} F(x,v) dx \nonumber \\
&\leq \frac{r^{2}}{2}-A|v(\cdot,0)|_{L^{\mu}(0,T)^{N}}^{\mu}+BT^{N} \nonumber \\
&\leq \frac{r^{2}}{2}-C''(m^{2s}c^{2}T^{N}+r^{2})^{\frac{\mu}{2}} +BT^{N} \label{4.12} \\
&\leq (m^{2s}c^{2}T^{N}+r^{2})-C''(m^{2s}c^{2}T^{N}+r^{2})^{\frac{\mu}{2}} +BT^{N} \label{4.13}\\
&=||v||_{\mathbb{X}^{s}_{m,T}}^{2}-E||v||_{\mathbb{X}^{s}_{m,T}}^{\mu}+F. \label{4.14}
\end{align}

We recall that $\mu>2$.
By using (\ref{4.12}), there exists $R>0$ such that
$$
\mathcal{J}_{m}(y+rz) \leq 0 \mbox{ for any } r \geq  R \mbox{ and } y \in \mathbb{Y}^{s}_{m,T}.
$$
Let $r\in [0,R]$. By (\ref{4.13}), we can find $R'>0$ such that $\mathcal{J}_{m}(y+rz) \leq 0$ for $||y||_{\mathbb{X}^{s}_{m,T}} \geq R'$. By (\ref{4.14}), we deduce that there exists a constant $\delta>0$ such that $\mathcal{J}_{m}(v)\leq \delta$ for any $v\in \mathbb{A}^{s}_{m,T}$.

\end{proof}

Finally we show that $\mathcal{J}_{m}$ satisfies the Palais-Smale condition:

\begin{lem}\label{Linking4}
Let $c \in \mathbb{R}$. Let $(v_{j})\subset \mathbb{X}^{s}_{m,T}$ be a sequence such that
\begin{equation}\label{PS}
\mathcal{J}_{m}(v_{j}) \rightarrow c \mbox{ and }  \mathcal{J}_{m}'(v_{j}) \rightarrow 0.
\end{equation}
Then there exist a subsequence $(v_{j_{h}})\subset (v_{j})$ and $v \in \mathbb{X}^{s}_{m,T}$ such that $v_{j_{h}} \rightarrow v$ in $\mathbb{X}^{s}_{m,T}$.
\end{lem}

\begin{proof}
We start proving that $(v_{j})$ is bounded in $\mathbb{X}^{s}_{m,T}$.

Fix $\beta \in (\frac{1}{\mu}, \frac{1}{2})$.
By using Lemma \ref{lfF} with $\varepsilon=1$, we get
\begin{align}\label{teresa1}
\Bigl|\int_{\partial^{0}\mathcal{S}_{T}\cap \{|v_{j}|\leq r_{0}\}} \beta f(x,v_{j})v_{j}-F(x,v_{j}) dx \Bigr| &\leq ((2\beta +1) r_{0}^{2} + C_{1} (p+2) r_{0}^{p+1}) T^{N}= \iota_{1}
\end{align}
and
\begin{align}\label{teresa2}
\Bigl|\int_{\partial^{0}\mathcal{S}_{T}\cap \{|v_{j}|\leq r_{0}\}} F(x,v_{j}) dx \Bigr| &\leq (r_{0}^{2}+ C_{1} r_{0}^{p+1}) T^{N}= \iota_{2}.
\end{align}
Taking into account Lemma \ref{F**}, $(f5)$, (\ref{partequad}), (\ref{PS}), (\ref{teresa1}) and (\ref{teresa2}) we have for $j$ large
\begin{align}
&c+1+||v_{j}||_{\mathbb{X}^{s}_{m,T}} \geq  \mathcal{J}_{m}(v_{j})-\beta \langle  \mathcal{J}'(v_{j}), v_{j} \rangle \nonumber \\
&=\Bigl(\frac{1}{2}-\beta \Bigr) \Bigl[ ||v_{j}||_{\mathbb{X}^{s}_{m,T}}^{2}-m^{2s}|v_{j}(\cdot,0)|_{L^{2}(0,T)^{N}}^{2} \Bigr]+\int_{\partial^{0}\mathcal{S}_{T}} \beta f(x,v_{j})v_{j}-F(x,v_{j}) dx \nonumber \\
&\geq \int_{\partial^{0} \mathcal{S}_{T}} [\beta f(x,v_{j})v_{j}-F(x,v_{j})]dx \nonumber \\
&= \int_{\partial^{0} \mathcal{S}_{T}\cap \{|v_{n}|\geq r_{0}\}} [\beta f(x,v_{j})v_{j}-F(x,v_{j})]dx  +  \int_{\partial^{0} \mathcal{S}_{T}\cap \{|v_{n}|\leq r_{0}\}} [\beta f(x,v_{j})v_{j}-F(x,v_{j})]dx  \nonumber \\
&\geq (\mu \beta-1) \int_{\partial^{0}\mathcal{S}_{T}\cap \{|v_{j}|\geq r_{0}\}} F(x,v_{j}) dx-\iota_{1} \nonumber\\
&\geq (\mu \beta -1)\int_{\partial^{0} \mathcal{S}_{T}} F(x,v_{j})dx -(\mu \beta -1)\iota_{2}  -\iota_{1} \nonumber \\
&= (\mu \beta-1) \int_{\partial^{0}\mathcal{S}_{T}} F(x,v_{j}) dx-\iota \label{4.2}\\
& \geq (\mu \beta-1)[a_{3}|v_{j}(\cdot,0)|_{L^{\mu}(0,T)^{N}}^{\mu}-a_{4}T^{N}]-\iota \nonumber \\
& \geq (\mu \beta-1)[a_{3}|v_{j}(\cdot,0)|^{\mu}_{L^{2}(0,T)^{N}}T^{-N\frac{\mu-2}{2}}-a_{4}T^{N}]-\iota. \label{4.3}
\end{align}

Hence, by (\ref{4.2}) and (\ref{4.3}), we deduce that
\begin{align*}
||v_{j}||_{\mathbb{X}^{s}_{m,T}}^{2} &=2 \mathcal{J}_{m}(v_{j})+m^{2s}|v_{j}(\cdot,0)|^{2}_{L^{2}(0,T)^{N}}+2\int_{\partial^{0}\mathcal{S}_{T}} F(x,v_{j}) dx \\
& \leq C_{1}+C_{2} (C_{3}+1+||v_{j}||_{\mathbb{X}^{s}_{m,T}})^{\frac{2}{\mu}}+ C_{4}(C_{5} +1+||v_{j}||_{\mathbb{X}^{s}_{m,T}}) \\
& \leq C_{6}+ C_{7} ||v_{j}||_{\mathbb{X}^{s}_{m,T}}
\end{align*}
that is $(v_{j})$ is bounded in $\mathbb{X}^{s}_{m,T}$.

By Theorem \ref{tracethm} we can assume, up to a subsequence,  that
\begin{align}\label{3.7}
v_{j} &\rightharpoonup v \mbox{ in } \mathbb{X}^{s}_{m,T}  \nonumber \\
v_{j}(\cdot,0) &\rightarrow v(\cdot,0) \mbox{ in } L^{p+1}(0,T)^{N} \\
v_{j}(\cdot,0) &\rightarrow  v(\cdot,0) \mbox{ a.e in } (0,T)^{N} \nonumber
\end{align}
as $j\rightarrow \infty$ and there exists $h\in L^{p+1}(0,T)^{N}$ such that
\begin{equation}\label{3.10}
|v_{j}(x,0)|\leq h(x) \quad \mbox{ a.e. in } x\in (0,T)^{N}, \quad \mbox{ for all } j\in \mathbb{N}.
\end{equation}
Taking into account $(f2)$, $(f4)$, (\ref{3.7}), (\ref{3.10}) and the Dominated Convergence Theorem  we get
\begin{equation}\label{1}
\int_{\partial^{0}\mathcal{S}_{T}} f(x,v_{j}) v_{j} dx \rightarrow \int_{\partial^{0}\mathcal{S}_{T}} f(x,v) v dx
\end{equation}
and
\begin{equation}\label{2}
\int_{\partial^{0}\mathcal{S}_{T}} f(x,v_{j}) v dx \rightarrow \int_{\partial^{0}\mathcal{S}_{T}} f(x,v) v dx
\end{equation}
as $j \rightarrow \infty$.

By using (\ref{PS}) and the boundedness of $(v_{j})_{j\in \mathbb{N}}$ in $\mathbb{X}^{s}_{m,T}$, we deduce that
$\langle  \mathcal{J}_{m}'(v_{j}),v_{j} \rangle \rightarrow 0$, that is
\begin{equation}\label{wc1}
||v_{j}||_{\mathbb{X}^{s}_{m,T}}^{2}- m^{2s}|v_{j}(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}-\int_{\partial^{0}\mathcal{S}_{T}} f(x,v_{j}) v_{j} dx \rightarrow 0
\end{equation}
as $j\rightarrow \infty$.
By $(\ref{3.7})$, (\ref{1}) and (\ref{wc1}) we have
\begin{equation}\label{wc2}
||v_{j}||_{\mathbb{X}^{s}_{m,T}}^{2} \rightarrow m^{2s}|v(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}-\int_{\partial^{0}\mathcal{S}_{T}} f(x,v) v dx.
\end{equation}
Moreover, by (\ref{PS}) and $v \in \mathbb{X}^{s}_{m,T}$, we have $\langle  \mathcal{J}_{m}'(v_{j}),v \rangle \rightarrow 0$ as $j\rightarrow \infty$, that is
\begin{equation}\label{wc3}
\langle v_{j}, v \rangle_{\mathbb{X}^{s}_{m,T}}- m^{2s} \langle v_{j},v \rangle_{L^{2}(0,T)^{N}}-\int_{\partial^{0}\mathcal{S}_{T}} f(x,v_{j}) v dx \rightarrow 0
\end{equation}
Taking into account (\ref{3.7}), (\ref{3.10}), (\ref{2}) and (\ref{wc3}) we obtain
\begin{equation}\label{wc4}
||v||_{\mathbb{X}^{s}_{m,T}}^{2}= m^{2s}|v(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}-\int_{\partial^{0}\mathcal{S}_{T}} f(x,v) v dx.
\end{equation}
Thus, (\ref{wc2}) and (\ref{wc4}) imply that
\begin{equation}\label{wc5}
||v_{j}||_{\mathbb{X}^{s}_{m,T}}^{2}\rightarrow ||v||_{\mathbb{X}^{s}_{m,T}}^{2} \mbox{ as } j\rightarrow \infty.
\end{equation}
Since $\mathbb{X}^{s}_{m,T}$ is a Hilbert space, we have
$$
||v_{j}-v||_{\mathbb{X}^{s}_{m,T}}^{2}=||v_{j}||_{\mathbb{X}^{s}_{m,T}}^{2}+||v||_{\mathbb{X}^{s}_{m,T}}^{2}-2 \langle v_{j},v \rangle_{\mathbb{X}^{s}_{m,T}}
$$
and using $v_{j} \rightharpoonup v$ in $\mathbb{X}^{s}_{m,T}$ and (\ref{wc5})
we can conclude that $v_{j} \rightarrow v$ in $\mathbb{X}^{s}_{m,T}$, as $j \rightarrow \infty$.

\end{proof}

\begin{proof} [of Theorem \ref{thm1}]
Taking into account Lemma \ref{Linking1} - Lemma \ref{Linking4}, by the Theorem \ref{Linking Thm} we deduce that for any fixed $m>0$, there exists of a function $v\in \mathbb{X}^{s}_{m,T}$ such that
$$
\mathcal{J}_{m}(v_{m})=\alpha_{m}, \quad \mathcal{J}'_{m}(v_{m})=0
$$
where
\begin{equation}\label{criticalvalue}
\alpha_{m}=\inf_{\gamma \in \Gamma_{m}} \max_{v\in \mathbb{A}^{s}_{m,T}} \mathcal{J}_{m}(\gamma(v))
\end{equation}
and
$$
\Gamma_{m}=\{\gamma \in \mathcal{C}(\mathbb{A}^{s}_{m,T}, \mathbb{X}^{s}_{m,T}): \gamma=Id \mbox{ on } \partial \mathbb{A}^{s}_{m,T}\}.
$$

\end{proof}

\begin{remark}
Let us observe that an easy consequence of Theorem \ref{thm1} is the existence of infinitely many distinct $T$-periodic solutions to (\ref{P}). To prove it, one can proceed as in the proof of Corollary $6.44$ in \cite{Rab}.
\end{remark}

\section{Regularity of solutions to $(\ref{P})$}

In this section we study the regularity of weak solutions to the problem $(\ref{P})$.

Firstly we prove the following

\begin{lem}\label{lemmino}

Let $v\in \mathbb{X}^{s}_{m,T}$ be a weak solution to
\begin{equation}
\left\{
\begin{array}{ll}
-\dive(y^{1-2s} \nabla v)+m^{2}y^{1-2s}v =0 &\mbox{ in }\mathcal{S}_{T} \\
v_{| {\{x_{i}=0\}}}= v_{| {\{x_{i}=T\}}} &\mbox{ on } \partial_{L}\mathcal{S}_{T} \\
\frac{\partial v}{\partial \nu^{1-2s}}=m^{2s}v+f(x,v)  &\mbox{ on } \partial^{0} \mathcal{S}_{T}
\end{array}.
\right.
\end{equation}
Then $v(\cdot,0)\in L^{q}(0,T)^{N}$ for all $q<\infty$.
\end{lem}
\begin{proof}
We proceed as in the proof of Lemma $7$ in \cite{A2}.
Since $v$ is a critical point for $\mathcal{J}_{m}$, we know that
\begin{equation}\label{criticpoint}
\iint_{\mathcal{S}_{T}} y^{1-2s}(\nabla v \nabla \eta+m^{2}v \eta) \, dxdy=\int_{\partial^{0} \mathcal{S}_{T}} m^{2s}v\eta+f(x,v)\eta \,dx
\end{equation}
for all $\eta\in \mathbb{X}^{m}_{T}$.

Let $w=vv^{2\beta}_{K}\in \mathbb{X}^{s}_{m,T}$ where $v_{K}=\min\{|v|,K\}$, $K>1$ and $\beta\geq 0$.
Taking $\eta=w$ in (\ref{criticpoint}) we deduce that
\begin{align}\label{conto1}
\iint_{\mathcal{S}_{T}}  &y^{1-2s}v^{2\beta}_{K}(|\nabla v|^{2}+m^{2}v^{2}) \, dxdy+\iint_{D_{K,T}} 2\beta y^{1-2s}v^{2\beta}_{K} |\nabla v|^{2} \, dx dy  \nonumber \\
&=m^{2s}\int_{\partial^{0}\mathcal{S}_{T}} v^{2} v^{2\beta}_{K} \,dx+ \int_{\partial^{0}\mathcal{S}_{T}} f(x,v)vv^{2\beta}_{K} \,dx
\end{align}
where $D_{K,T}=\{(x,y)\in \mathcal{S}_{T}: |v(x,y)|\leq K\}$. \\
It is easy to see that
\begin{align}\label{conto2}
\iint_{\mathcal{S}_{T}} &y^{1-2s}|\nabla (vv_{K}^{\beta})|^{2} dxdy \nonumber \\
&=\iint_{\mathcal{S}_{T}} y^{1-2s}v_{K}^{2\beta} |\nabla v|^{2} dxdy+\iint_{D_{K,T}} (2\beta+\beta^{2}) y^{1-2s}v_{K}^{2\beta} |\nabla v|^{2} dxdy.
\end{align}
Then, putting together (\ref{conto1}) and (\ref{conto2}) we get
\begin{align}\label{S1}
&||vv_{K}^{\beta}||_{\mathbb{X}^{s}_{m,T}}^{2} \nonumber \\
&=\iint_{\mathcal{S}_{T}} y^{1-2s}[|\nabla (vv_{K}^{\beta})|^{2}+m^{2}v^{2}v_{K}^{2\beta}] dxdy \nonumber \\
&=\iint_{\mathcal{S}_{T}} y^{1-2s}v_{K}^{2\beta}[ |\nabla v|^{2}+m^{2}v^{2}] dxdy+\iint_{D_{K,T}} 2\beta \Bigl(1+\frac{\beta}{2}\Bigr) y^{1-2s}v_{K}^{2\beta} |\nabla v|^{2} dxdy \nonumber \\
&\leq c_{\beta} \Bigl[\iint_{\mathcal{S}_{T}} y^{1-2s}v_{K}^{2\beta}[ |\nabla v|^{2}+m^{2}v^{2}] dxdy+\iint_{D_{K,T}} 2\beta y^{1-2s}v_{K}^{2\beta} |\nabla v|^{2} dxdy\Bigr] \nonumber \\
&=c_{\beta} \int_{\partial^{0}\mathcal{S}_{T}} m^{2s}v^{2}v_{K}^{2\beta} + f(x,v)v v_{K}^{2\beta} \,dx
\end{align}
where $c_{\beta}=1+\frac{\beta}{2}$.
By Lemma \ref{lfF} with $\varepsilon=1$ we deduce that
\begin{equation*}
m^{2s}v^{2}v_{K}^{2\beta} + f(x,v)v v_{K}^{2\beta}\leq (m^{2s}+2)v^{2}v_{K}^{2\beta}+(p+1)C_{1}|v|^{p-1}v^{2}v_{K}^{2\beta} \mbox{ on } \partial^{0}\mathcal{S}_{T}.
\end{equation*}
Now, we prove that
\begin{equation*}
|v|^{p-1}\leq 1+h \mbox{ on } \partial^{0}\mathcal{S}_{T}
\end{equation*}
for some $h\in L^{N/2s}(0,T)^{N}$.
Firstly, we observe that
$$
|v|^{p-1}=\chi_{\{|v|\leq 1\}}|v|^{p-1}+\chi_{\{|v|>1\}}|v|^{p-1}\leq 1+\chi_{\{|v|>1\}}|v|^{p-1}\ \mbox{ on } \partial^{0}\mathcal{S}_{T}.
$$
If $(p-1)N<4s$ then
$$
\int_{\partial^{0}\mathcal{S}_{T}} \chi_{\{|v|>1\}}|v|^{\frac{N}{2s}(p-1)} dx \leq \int_{\partial^{0}\mathcal{S}_{T}} \chi_{\{|v|>1\}}|v|^{2} dx<\infty
$$
while if $4s\leq (p-1)N$ we have that $(p-1)\frac{N}{2s}\in [2,\frac{2N}{N-2s}]$.

Therefore, there exist a constant $c=m^{2s}+2+(p+1)C_{1}$ and a function $h\in L^{N/2s}(0,T)^{N}$, $h\geq 0$ and independent of $K$ and $\beta$, such that
\begin{equation}\label{S2}
m^{2s}v^{2}v_{K}^{2\beta} + f(x,v)vv_{K}^{2\beta}\leq (c+h)v^{2}v_{K}^{2\beta} \mbox{ on } \partial^{0}\mathcal{S}_{T}.
\end{equation}
Taking into account (\ref{S1}) and (\ref{S2}) we have
\begin{equation*}
||vv_{K}^{\beta}||_{\mathbb{X}^{s}_{m,T}}^{2}\leq c_{\beta} \int_{\partial^{0}\mathcal{S}_{T}} (c+h)v^{2}v_{K}^{2\beta} dx,
\end{equation*}
and by Monotone Convergence Theorem ($v_{K}$ is increasing with respect to $K$) we have as $K\rightarrow \infty$
\begin{equation}\label{i1}
|||v|^{\beta+1}||_{\mathbb{X}^{s}_{m,T}}^{2}\leq cc_{\beta} \int_{\partial^{0}\mathcal{S}_{T}} |v|^{2(\beta +1)} dx + c_{\beta}\int_{\partial^{0}\mathcal{S}_{T}}  h|v|^{2(\beta +1)}dx.
\end{equation}
Fix $M>0$ and let $A_{1}=\{h\leq M\}$ and $A_{2}=\{h>M\}$.

Then
\begin{equation}\label{i2}
\int_{\partial^{0}\mathcal{S}_{T}}  h|v(\cdot,0)|^{2(\beta +1)} dx\leq M||v(\cdot,0)|^{\beta+1}|_{L^{2}(0,T)^{N}}^{2}+\varepsilon(M)||v(\cdot,0)|^{\beta+1}|_{L^{2^{\sharp}}(0,T)^{N}}^{2}
\end{equation}
where $\varepsilon(M)=\Bigl(\int_{A_{2}} h^{N/2s} dx \Bigr)^{\frac{2s}{N}}\rightarrow 0$ as $M\rightarrow \infty$.
Taking into account (\ref{i1}), (\ref{i2}), we get
\begin{equation}\label{regv}
|||v|^{\beta+1}||_{\mathbb{X}^{s}_{m,T}}^{2}\leq c_{\beta}(c+M)||v(\cdot,0)|^{\beta+1}|_{L^{2}(0,T)^{N}}^{2}+c_{\beta}\varepsilon(M)||v(\cdot,0)|^{\beta+1}|_{L^{2^{\sharp}_{s}}(0,T)^{N}}^{2}.
\end{equation}
By using Theorem $\ref{thm2}$ we know that there exists a constant $C^{2}_{2^{\sharp}_{s},m}>0$ such that
\begin{equation}\label{S3}
||v(\cdot,0)|^{\beta+1}|_{L^{2^{\sharp}_{s}}(0,T)^{N}}^{2}\leq C^{2}_{2^{\sharp}_{s},m} |||v|^{\beta+1}||_{\mathbb{X}^{s}_{m,T}}^{2}.
\end{equation}
Then, choosing $M$ large so that $\varepsilon(M) c_{\beta} C^{2}_{2^{\sharp},m}<\frac{1}{2}$,
and by using $(\ref{regv})$ and $(\ref{S3})$ we obtain
\begin{equation}\label{iter}
||v(\cdot,0)|^{\beta+1}|_{L^{2^{\sharp}_{s}}(0,T)^{N}}^{2}\leq 2 C^{2}_{2^{\sharp}_{s},m} c_{\beta}(c+M)||v(\cdot,0)|^{\beta+1}|^{2}_{L^{2}(0,T)^{N}}.
\end{equation}
Then we can start a bootstrap argument: since $v(\cdot,0)\in L^{\frac{2N}{N-2s}}$ we can apply (\ref{iter}) with $\beta_{1}+1=\frac{N}{N-2s}$ to deduce that $v(\cdot,0)\in L^{\frac{(\beta_{1}+1)2N}{N-2s}}(0,T)^{N}=L^{\frac{2N^{2}}{(N-2s)^{2}}}(0,T)^{N}$. Applying (\ref{iter}) again, after $k$ iterations, we find $v(\cdot,0)\in L^{\frac{2N^{k}}{(N-2s)^{k}}}(0,T)^{N}$, and so $v(\cdot,0)\in L^{q}(0,T)^{N}$ for all $q\in[2,\infty)$.

\end{proof}

Then, we can deduce the following result:

\begin{thm}\label{regularity}
Let $v\in \mathbb{X}^{s}_{m,T}$ be a weak solution to
\begin{equation}
\left\{
\begin{array}{ll}
-\dive(y^{1-2s} \nabla v)+m^{2}y^{1-2s}v =0 &\mbox{ in }\mathcal{S}_{T} \\
v_{| {\{x_{i}=0\}}}= v_{| {\{x_{i}=T\}}} &\mbox{ on } \partial_{L}\mathcal{S}_{T} \\
\frac{\partial v}{\partial \nu^{1-2s}}=\kappa_{s}[m^{2s}v+f(x,v)]  &\mbox{ on } \partial^{0} \mathcal{S}_{T}
\end{array}.
\right.
\end{equation}
Let us assume that $v$ is extended by periodicity to the whole $\mathbb{R}^{N+1}_{+}$.
Then $v(\cdot,0)\in \mathcal{C}^{0,\alpha}(\mathbb{R}^{N})$ for some $\alpha \in (0,1)$.
\end{thm}
\begin{proof}
It is clear that $v \in H^{1}_{m}(A\times \mathbb{R}_{+},y^{1-2s})$ for any bounded domain $A\subset \mathbb{R}^{N}$.
By using Lemma \ref{lemmino} here and Proposition $3.5.$  in \cite{FallFelli} we deduce the thesis.

\end{proof}

\section{Passage to the limit as $m \rightarrow 0$}

In this last section, we give the proof of Theorem \ref{thmdue}. We verify that it is possible to take the limit in (\ref{R}) as $m\rightarrow 0$ so that we deduce the existence of a nontrivial weak solution to (\ref{P'}).
In particular, we will prove that such solution is H\"older continuous.
We remark that in Section $4$ we proved that for any $m>0$ there exists $v_{m}\in \mathbb{X}^{s}_{m,T}$ such that
\begin{align}\label{zero}
\mathcal{J}_{m}(v_{m})= \alpha_{m} \quad \mbox{ and } \quad \mathcal{J}'_{m}(v_{m})=0,
\end{align}
where $\alpha_{m}$ is defined in (\ref{criticalvalue}).
In order to attain our aim, we estimate, from above and below, the critical levels of the functional $\mathcal{J}_{m}$ independently of $m$.

Let us assume that $0<m<m_{0}:=\frac{1}{2C^{2}_{2^{\sharp}_{s}}}$, where $C_{2^{\sharp}_{s}}$ is the Sobolev constant which appears in (\ref{benyi}).

We start proving that there exists a positive constant $\delta$ independent of $m$ such that
\begin{equation}\label{cmdelta}
\alpha_{m}\leq \delta \quad \mbox{ for all }  0<m<m_{0}.
\end{equation}

By using (\ref{einstein}) and $m<m_{0}$ we know that
$$
C_{1}\leq ||z||_{\mathbb{X}^{s}_{m,T}}^{2} \leq C_{2}+m_{0}^{2}C_{3}.
$$
Moreover (see Lemma \ref{lemma5}) we have for any $v=y+rz \in \mathbb{Y}^{s}_{m,T} \oplus \mathbb{R}_{+}z$
\begin{align}
|v(\cdot,0)|_{L^{\mu}(0,T)^{N}}^{\mu} &\geq T^{-\frac{N(\mu-2)}{2}}|v(\cdot,0)|_{L^{2}(0,T)^{N}}^{\mu} \nonumber \\
&=T^{-\frac{N(\mu-2)}{2}}\Bigl(\int_{(0,T)^{N}} (c+rz)^{2} dx \Bigr)^{\frac{\mu}{2}} \nonumber \\
& \geq  T^{-\frac{N(\mu-2)}{2}} \Bigl(c^{2}T^{N}+(T/2)^{N}\frac{r^{2}}{||z||^{2}_{\mathbb{X}^{s}_{m,T}}} \Bigr)^{\frac{\mu}{2}} \nonumber\\
& \geq  T^{-\frac{N(\mu-2)}{2}} \min\Bigl \{\frac{1}{m^{2s}_{0}}, \frac{(T/2)^{N}}{C_{2}+m_{0}^{2}C_{3}} \Bigr\}(m^{2s}c^{2}T^{N}+r^{2})^{\frac{\mu}{2}} \nonumber \\
&= C||v||_{\mathbb{X}^{s}_{m,T}}^{\mu}
\end{align}
for some $C=C(m_{0},T,N,s)>0$.

Then, for any $v=y+rz\in \mathbb{Y}^{s}_{m,T}\oplus \mathbb{R}_{+} z$ and $0<m<m_{0}$ we get
\begin{align}
\mathcal{J}_{m}(v)&=\frac{1}{2}||v||_{\mathbb{X}^{s}_{m,T}}^{2}-\frac{m^{2s}}{2}|v(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}-\int_{\partial^{0} \mathcal{S}_{T}} F(x,v) dx \nonumber\\
&\leq \frac{1}{2}||v||_{\mathbb{X}^{s}_{m,T}}^{2} -A|v(\cdot,0)|_{L^{\mu}(0,T)^{N}}^{\mu}+B\, T^{N} \nonumber \\
&=||v||_{\mathbb{X}^{s}_{m,T}}^{2}-C||v||_{\mathbb{X}^{s}_{m,T}}^{\mu}+D\leq \delta
\end{align}
where $A, B, C, D, \delta>0$ are independent of $m$.

Now we prove that there exists $\lambda>0$ independent of $m$, such that
\begin{equation}\label{cmlambda}
\alpha_{m}\geq \lambda \quad \mbox{ for all }  0<m<m_{0}.
\end{equation}

Let $v\in \mathbb{Z}^{s}_{m,T}$ and $\varepsilon>0$. We denote by $c_{k}$ the Fourier coefficients of the trace of $v$.

By using (\ref{benyi}) and (\ref{eqYY}) (with $\kappa_{s}=1$), we know that
\begin{align}
|v|_{L^{q}(0,T)^{N}} &\leq C_{2^{\sharp}_{s}} \Bigl( \sum_{|k|\geq 1} \omega ^{2s} |k|^{2s} |c_{k}|^{2} \Bigr)^{\frac{1}{2}} \nonumber \\
&\leq C_{2^{\sharp}_{s}}  |v|_{\mathbb{H}^{s}_{m,T}} \nonumber \\
&\leq C_{2^{\sharp}_{s}}  ||v||_{\mathbb{X}^{s}_{m,T}} \label{benyioh}
\end{align}
for any $q\in [2, 2^{\sharp}_{s}]$.

By using Lemma \ref{lfF} and (\ref{benyioh}) we can see that for every $0<m<m_{0}$

\begin{align*}
\mathcal{J}_{m}(v)&=\frac{1}{2}\iint_{\mathcal{S}_{T}} y^{1-2s}(|\nabla v|^{2}+m^{2} v^{2}) \ dx dy-\frac{m^{2s}}{2}\int_{\partial^{0} \mathcal{S}_{T}} |v|^{2} dx -\int_{\partial^{0} \mathcal{S}_{T}} F(x,v) dx \\
&\geq \frac{1}{2} ||v||^{2}_{\mathbb{X}^{s}_{m,T}}-\Bigl(\frac{m}{2}+\varepsilon \Bigr) |v(\cdot,0)|_{L^{2}(0,T)^{N}}^{2} -C_{\varepsilon} |v(\cdot,0)|_{L^{p+1}(0,T)^{N}}^{p+1} \\
&\geq  \Bigl[\frac{1}{2}-C^{2}_{2^{\sharp}_{s}} \Bigl(\frac{m}{2}+\varepsilon \Bigr) \Bigr] ||v||^{2}_{\mathbb{X}^{m}_{T}} -C_{\varepsilon} C^{p+1}_{2^{\sharp}_{s}} ||v||_{\mathbb{X}^{s}_{m,T}}^{p+1}    \\
& \geq \Bigl(\frac{1}{4}- C^{2}_{2^{\sharp}_{s}}\varepsilon   \Bigr) ||v||^{2}_{\mathbb{X}^{s}_{m,T}} -C'_{\varepsilon}||v||_{\mathbb{X}^{s}_{m,T}}^{p+1} .
\end{align*}

Choosing $0<\varepsilon<\frac{1}{4 C^{2}_{2^{\sharp}_{s}}}$, we have that $b:=\frac{1}{4}-C^{2}_{2^{\sharp}_{s}}\varepsilon>0$.\\
Let $\rho:=\Bigl(\frac{b}{2C'_{b}}\Bigr)^{\frac{1}{p-1}}$.  Then, for every $v\in \mathbb{Z}^{s}_{m,T}$ such that $||v||_{\mathbb{X}_{m,T}^{s}}=\rho$
$$
\mathcal{J}_{m}(v) \geq  b\rho^{2}-C'_{b}\, \rho^{p+1}= \frac{b}{2} \Bigl(\frac{b}{2C'_{b}} \Bigr)^{\frac{2}{p-1}}=:\lambda .
$$
Therefore, taking into account (\ref{cmdelta}) and (\ref{cmlambda}) we deduce that
\begin{equation}\label{alpham}
\lambda \leq \alpha_{m}\leq \delta \,  \mbox{ for every } \,  0<m<m_{0}.
\end{equation}

Now, we estimate the $H^{1}_{loc}(\mathcal{S}_{T}, y^{1-2s})$- norm of $v_{m}$ in order to pass to the limit in $(\ref{R})$ as $m \rightarrow 0$.

Fix $\beta \in (\frac{1}{\mu},\frac{1}{2})$.
By using  (\ref{zero}) and (\ref{alpham}), we have for any $m\in (0, m_{0})$
\begin{align}
\delta &\geq \mathcal{J}_{m}(v_{m})-\beta \langle \mathcal{J}_{m}'(v_{m}),v_{m} \rangle  \nonumber \\
&=\Bigl(\frac{1}{2}-\beta\Bigr) [ ||v_{m}||_{\mathbb{X}^{s}_{m,T}}^{2}- m^{2s}|v_{m}(\cdot,0)|_{L^{2}(0,T)^{N}}^{2} ] \\
&+\int_{\partial^{0} \mathcal{S}_{T}} [\beta f(x,v_{m})v_{m}-F(x,v_{m})]dx  \nonumber \\
&\geq \int_{\partial^{0} \mathcal{S}_{T}} [\beta f(x,v_{m})v_{m}-F(x,v_{m})]dx  \nonumber \\
&\geq (\mu \beta -1)\int_{\partial^{0} \mathcal{S}_{T}} F(x,v_{m})dx -\tilde{\kappa}   \label{Fineq} \\
&\geq (\mu \beta-1)[a_{3}|v_{m}(\cdot,0)|^{\mu}_{L^{\mu}(0,T)^{N}}-a_{4}T^{N}]-\tilde{\kappa} \nonumber  \\
&\geq (\mu \beta-1)[a_{3}|v_{m}(\cdot,0)|^{\mu}_{L^{2}(0,T)^{N}}T^{-N\frac{\mu-2}{2}}-a_{4}T^{N}]-\tilde{\kappa}. \label{Fi}
\end{align}
By (\ref{Fi}) we deduce that the trace of $v_{m}$ is bounded in $L^{2}(0,T)^{N}$
\begin{equation}\label{vti}
|v_{m}(\cdot,0)|_{L^{2}(0,T)^{N}} \leq K(\delta) \mbox{ for every } m\in (0,m_{0}).
\end{equation}
Taking into account (\ref{zero}), (\ref{alpham}), (\ref{Fineq}) and (\ref{vti}) we deduce
\begin{align}\label{nablavti}
||\nabla v_{m}||^{2}_{L^{2}(\mathcal{S}_{T},y^{1-2s})}&\leq ||v_{m}||^{2}_{\mathbb{X}^{s}_{m,T}} \nonumber \\
&=2J_{m}(v_{m})+m|v_{m}(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}+2\int_{\partial^{0} \mathcal{S}_{T}} F(x,v_{m})dx \nonumber  \\
&\leq 2\delta+\frac{\omega}{2}K(\delta)+C(\delta)=:K'(\delta).
\end{align}
Now, let  $c_{k}^{m}$ be the Fourier coefficients of the trace of $v_{m}$.
By (\ref{eqYY}) we can see that
\begin{align}\label{compik}
K'(\delta)&\geq ||v_{m}||^{2}_{\mathbb{X}^{s}_{m,T}}\geq |v_{m}(\cdot,0)|^{2}_{\mathbb{H}^{s}_{m,T}}\nonumber \\
&\geq \sum_{k\in \mathbb{Z}^{N}} \omega^{2s}|k|^{2s}|c_{k}^{m}|^{2},
\end{align}
which, together with (\ref{vti}), implies that
\begin{align}\label{baroni}
|v_{m}(\cdot,0)|_{\mathbb{H}^{s}_{T}}\leq K''(\delta) \mbox{ for every } m\in (0,m_{0})
\end{align}
that is $\textup{Tr}(v_{m})$ is bounded in $\mathbb{H}^{s}_{T}$.

Finally we estimate the $L^{2}_{loc}(\mathcal{S}_{T},y^{1-2s})$-norm of $v_{m}$ uniformly in $m$.

Fix $\alpha>0$ and let $v\in C^{\infty}_{T}(\overline{\mathbb{R}^{N+1}_{+}})$ such that $||v_{m}||_{\mathbb{X}^{s}_{m,T}}<\infty$.
For any $x\in [0,T]^{N}$ and $y\in [0, \alpha]$, we have
$$
v(x,y)=v(x,0)+\int_{0}^{y} \partial_{y} v(x,t) dt.
$$
By using $(a+b)^{2}\leq 2a^{2}+2b^{2}$ for all $a, b\geq 0$ we obtain
$$
|v(x,y)|^2 \leq 2  |v(x,0)|^{2}+2\Bigl(\int_{0}^{y}|\partial_{y} v(x,t)| dt\Bigr)^{2},
$$
and applying the H\"older inequality we deduce
\begin{equation}\label{vtii5}
|v(x,y)|^2 \leq 2 \Bigl[ |v(x,0)|^{2}+\Bigl(\int_{0}^{y} t^{1-2s}|\partial_{y} v(x,t)|^{2}dt\Bigr)\frac{y^{2s}}{2s}\,  \Bigr].
\end{equation}
Multiplying both members by $y^{1-2s}$ we have
\begin{equation}\label{vtii}
y^{1-2s}|v(x,y)|^2 \leq 2 \Bigl[ y^{1-2s}|v(x,0)|^{2}+\Bigl(\int_{0}^{y} t^{1-2s} |\partial_{y} v(x,t)|^{2}dt\Bigr)\frac{y}{2s} \Bigr].
\end{equation}
Integrating (\ref{vtii}) over $(0,T)^{N}\times (0,\alpha)$ we have
\begin{align}\label{nash}
||v||_{L^{2}((0,T)^{N}\times (0,\alpha),y^{1-2s})}^2 &\leq \frac{\alpha^{2-2s}}{1-s} |v(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}+\frac{\alpha ^{2}}{2s} ||\partial_{y} v||_{L^{2}(\mathcal{S}_{T},y^{1-2s})}^{2}.
\end{align}
By density, the above inequality holds for any $v\in \mathbb{X}^{s}_{m,T}$.

Then, by using (\ref{nash}) and exploiting (\ref{vti}) and (\ref{nablavti}) we have
\begin{align*}
||v_{m}||_{L^{2}((0,T)^{N}\times (0,\alpha),y^{1-2s})}^2 &\leq \frac{\alpha^{2-2s}}{1-s} |v_{m}(\cdot,0)|_{L^{2}(0,T)^{N}}^{2}+\frac{\alpha ^{2}}{2s} ||\partial_{y} v_{m}||_{L^{2}(\mathcal{S}_{T},y^{1-2s})}^{2} \\
& \leq C(\alpha,s) K(\delta)^{2}+C'(\alpha,s) K'(\delta)
\end{align*}
for any $0<m<m_{0}$.

As a consequence, we can extract a subsequence, that for simplicity we will denote again with  $(v_{m})$, and a function $v$ such that
\begin{itemize}
\item $v\in L^{2}_{loc}(\mathcal{S}_{T},y^{1-2s})$ and $\nabla v\in L^{2}(\mathcal{S}_{T},y^{1-2s})$;
\item $v_{m}\rightharpoonup v$ in $L^{2}_{loc}(\mathcal{S}_{T},y^{1-2s})$ as $m\rightarrow 0$;
\item $\nabla v_{m}\rightharpoonup \nabla v$ in $L^{2}(\mathcal{S}_{T},y^{1-2s})$ as $m\rightarrow 0$;
\item $v_{m}(\cdot,0)\rightharpoonup v(\cdot,0)$  in $\mathbb{H}^{s}_{T}$ and $v_{m}(\cdot,0)\rightarrow v(\cdot,0)$ in $L^{q}(0,T)^{N}$ as $m\rightarrow 0$, for any $q\in [2, \frac{2N}{N-2s})$.
\end{itemize}

Now we prove that $v$ is a weak solution to
\begin{equation}\label{Ros}
\left\{
\begin{array}{ll}
-\dive(y^{1-2s} \nabla v) =0 &\mbox{ in }\mathcal{S}_{T}:=(0,T)^{N} \times (0,\infty)  \\
v_{| {\{x_{i}=0\}}}= v_{| {\{x_{i}=T\}}} & \mbox{ on } \partial_{L}\mathcal{S}_{T}:=\partial (0,T)^{N} \times [0,\infty) \\
\frac{\partial v}{\partial \nu^{1-2s}}=f(x,v)   &\mbox{ on }\partial^{0}\mathcal{S}_{T}:=(0,T)^{N} \times \{0\}
\end{array}.
\right.
\end{equation}

Fix $\varphi \in \mathbb{X}^{s}_{T}$.
We know that $v_{m}$ satisfies
\begin{equation}\label{spqr}
\iint_{\mathcal{S}_{T}} y^{1-2s}(\nabla v_{m} \nabla \eta+m^{2}v_{m}\eta) \; dxdy=\int_{\partial^{0} \mathcal{S}_{T}} [m^{2s}v_{m}+f(x,v_{m})]\eta  \; dx
\end{equation}
for every $\eta \in \mathbb{X}^{s}_{m,T}$.
Now, we consider $\xi\in \mathcal{C}^{\infty}([0,\infty))$ defined as follows
\begin{equation}\label{xidef}
\left\{
\begin{array}{cc}
\xi=1 &\mbox{ if } 0\leq y\leq 1 \\
0\leq \xi \leq 1 &\mbox{ if } 1\leq y\leq 2 \\
\xi=0 &\mbox{ if } y\geq 2
\end{array}.
\right.
\end{equation}
We set $\xi_{R}(y)=\xi(\frac{y}{R})$ for $R>1$. Then choosing $\eta=\varphi \xi_{R}\in \mathbb{X}^{s}_{m,T}$ in (\ref{spqr}) and taking the limit as $m\rightarrow 0$ we have
\begin{equation}
\iint_{\mathcal{S}_{T}} y^{1-2s}\nabla v \nabla (\varphi \xi_{R}) \; dxdy=\int_{\partial^{0} \mathcal{S}_{T}} f(x,v)\varphi \; dx.
\end{equation}
Taking the limit as $R\rightarrow \infty$ we deduce that $v$ verifies
$$
\iint_{\mathcal{S}_{T}} y^{1-2s} \nabla v \nabla \varphi \; dxdy-\int_{\partial^{0} \mathcal{S}_{T}} f(x,v)\varphi  \; dx=0 \quad \forall \varphi \in \mathbb{X}^{s}_{T}.
$$
Now we want to prove that $v\not \equiv 0$.
Let $\xi \in \mathcal{C}^{\infty}([0,\infty))$ as in (\ref{xidef}) and we note that $\xi v \in \mathbb{X}^{s}_{m,T}$.

Then
\begin{align*}
0&= \langle \mathcal{J}'_{m}(v_{m}), \xi v \rangle \\
&= \iint_{{\mathcal{S}}_{T}} y^{1-2s}(\nabla v_{m} \nabla(\xi v) + m^{2} v_{m} \xi v) \, dxdy - m^{2s}\int_{\partial^{0} \mathcal{S}_{T}} v_{m} v \, dx \\
& - \int_{\partial^{0} \mathcal{S}_{T}} f(x,v_{m})v \,dx
\end{align*}
and taking the limit as $m\rightarrow 0$ we get
\begin{equation}\label{6.8}
0=\iint_{\mathcal{S}_{T}} y^{1-2s} \nabla v \nabla(\xi v) \, dxdy - \int_{\partial^{0} \mathcal{S}_{T}} f(x,v)v \, dx.
\end{equation}
By using the facts (\ref{zero}), (\ref{alpham}), $\langle \mathcal{J}'_{m}(v_{m}), v_{m}\rangle =0$ and $F\geq 0$, we have
\begin{align}\label{mF}
2\lambda &\leq 2\mathcal{J}_{m}(v_{m}) + m^{2s} |v_{m}(\cdot,0)|_{L^{2}(0,T)^{N}}^{2} + 2\int_{\partial^{0} \mathcal{S}_{T}} F(x, v_{m}) \, dx \nonumber \\
&=\|v_{m}\|_{\mathbb{X}^{m}_{T}}^{2} =m^{2s} |v_{m}(\cdot,0)|_{L^{2}(0,T)^{N}}^{2} + \int_{\partial^{0} \mathcal{S}_{T}} f(x,v_{m}) v_{m} \, dx.
\end{align}
Taking the limit in $(\ref{mF})$ as $m\rightarrow 0$ we obtain
\begin{equation}\label{6.9}
2\lambda \leq \int_{\partial^{0} \mathcal{S}_{T}} f(x, v)v \, dx.
\end{equation}
Hence, (\ref{6.8}) and (\ref{6.9}) give
\begin{equation*}
0<2\lambda \leq \int_{\partial^{0} \mathcal{S}_{T}} f(x, v)v \, dx = \iint_{\mathcal{S}_{T}} y^{1-2s}\nabla v \nabla(\xi v) \, dxdy.
\end{equation*}
that is $v$ is not a trivial solution to (\ref{Ros}).

Finally, we show that $v\in \mathcal{C}^{0,\alpha}([0,T]^{N})$, for some $\alpha \in (0,1)$.
We start proving that $v(\cdot,0)\in L^{q}(0,T)^{N}$ for any $q<\infty$.
We proceed as in the proof of Lemma \ref{lemmino} and we use the estimate (\ref{baroni}).

Let $w_{m}=v_{m}v^{2\beta}_{m,K}$ where $v_{m,K}=\min\{|v_{m}|,K\}$, $K>1$ and $\beta\geq 0$. Then, replacing $vv_{K}^{2\beta}$ by $v_{m}v^{2\beta}_{m,K}$ in (\ref{S1}), we can see that
\begin{align}
||v_{m}v_{m,K}^{\beta}||_{\mathbb{X}^{m}_{T}}^{2}\leq c_{\beta} \int_{(0,T)^{N}} [m^{2s}v_{m}^{2}v_{m,K}^{2\beta} + f(x,v_{m})v_{m} v_{m,K}^{2\beta}] dx
\end{align}
where $c_{\beta}=1+\frac{\beta}{2}\geq 1$.
Using Lemma \ref{lfF} with $\varepsilon=1$ we get
\begin{equation}
m^{2s}v_{m}^{2}v_{m,K}^{2\beta} + f(x,v_{m})v_{m} v_{m,K}^{2\beta}\leq (m^{2s}+2)v_{m}^{2}v_{m,K}^{2\beta}+(p+1)C_{1}|v_{m}|^{p-1}v^{2}v_{m,K}^{2\beta}.
\end{equation}
Since $v_{m}$ converges strongly in $L^{\frac{N(p-1)}{2s}}(0,T)^{N}$ (because of $\frac{N(p-1)}{2s}< 2^{\sharp}_{s}$), we can assume that, up to subsequences, there exists a function $z \in L^{\frac{N(p-1)}{2s}}(0,T)^{N}$ such that $|v_{m}(x,0)|\leq z(x)$ in $(0,T)^{N}$, for every $m<m_{0}$.

Therefore, there exist a constant $c=m^{2s}_{0}+2+(p+1)C_{1}$ and a function $h:=1+z^{p-1}\in L^{\frac{N}{2s}}(0,T)^{N}$, $h\geq 0$ and independent of $K$, $m$ and $\beta$ such that
\begin{equation}
m^{2s}v_{m}^{2}v_{m,K}^{2\beta} + f(x,v_{m})v_{m}v_{m,K}^{2\beta}\leq (c+h)v_{m}^{2}v_{m,K}^{2\beta} \mbox{ on } \partial^{0}\mathcal{S}_{T}.
\end{equation}
As a consequence
\begin{equation}
||v_{m}v_{m,K}^{\beta}||_{\mathbb{X}^{s}_{m,T}}^{2}\leq c_{\beta} \int_{(0,T)^{N}} (c+h)v_{m}^{2}v_{m,K}^{2\beta} dx.
\end{equation}
Taking the limit as $K\rightarrow \infty$ ($v_{m,K}$ is increasing with respect to $K$) we get
\begin{equation}\label{i177}
|||v_{m}|^{\beta+1}||_{\mathbb{X}^{s}_{m,T}}^{2}\leq cc_{\beta} \int_{(0,T)^{N}} |v_{m}|^{2(\beta +1)} + c_{\beta}\int_{(0,T)^{N}}  h|v_{m}|^{2(\beta +1)}dx.
\end{equation}
For any $M>0$, let $A_{1}=\{h\leq M\}$ and $A_{2}=\{h>M\}$.
Then
\begin{equation}\label{i277}
\int_{(0,T)^{N}}  h|v_{m}(\cdot,0)|^{2(\beta +1)}dx\leq M||v_{m}(\cdot,0)|^{\beta+1}|_{L^{2}(0,T)^{N}}^{2}+\varepsilon(M)||v_{m}(\cdot,0)|^{\beta+1}|_{L^{2^{\sharp}_{s}}(0,T)^{N}}^{2}
\end{equation}
where $\varepsilon(M)=\Bigl(\int_{A_{2}} h^{\frac{N}{2s}} dx \Bigr)^{\frac{2s}{N}}\rightarrow 0$ as $M\rightarrow \infty$.
Taking into account (\ref{i177}), (\ref{i277}), we have
\begin{equation}\label{regv77}
|||v_{m}|^{\beta+1}||_{\mathbb{X}^{s}_{m,T}}^{2}\leq c_{\beta}(c+M)||v_{m}(\cdot,0)|^{\beta+1}|_{L^{2}(0,T)^{N}}^{2}+c_{\beta}\varepsilon(M)||v_{m}(\cdot,0)|^{\beta+1}|_{L^{2^{\sharp}}(0,T)^{N}}^{2}.
\end{equation}
Now, by (\ref{benyi}), we know that for every $w\in \mathcal{C}^{\infty}_{T}(\mathbb{R}^{N})$ with mean zero, there exists $\mu_{0}:=C_{2^{\sharp}_{s}}>0$, such that
\begin{equation}\label{puffo}
|w|_{L^{2^{\sharp}_{s}}(0,T)^{N}}\leq \mu_{0} \Bigl(\sum_{|k|\neq 0} \omega^{2s}|k|^{2s} |b_{k}|^{2}\Bigr)^{1/2},
\end{equation}
where $b_{k}$ are the Fourier coefficients of $w$.

Therefore, if $w\in \mathcal{C}^{\infty}_{T}(\mathbb{R}^{N})$ and $\overline{w}:=\frac{1}{T^{N}}\int_{(0,T)^{N}} w(x)dx$, by using H\"older inequality we can see that
\begin{align}
|w|_{L^{2^{\sharp}_{s}}(0,T)^{N}}&\leq |w-\overline{w}|_{L^{2^{\sharp}_{s}}(0,T)^{N}}+|\overline{w}|_{L^{2^{\sharp}_{s}}(0,T)^{N}} \nonumber \\
&\leq \mu_{0} \Bigl(\sum_{|k|\neq 0} \omega^{2s}|k|^{2s} |b_{k}|^{2}\Bigr)^{1/2}+|\overline{w}|_{L^{2^{\sharp}_{s}}(0,T)^{N}} \nonumber  \\
&\leq \mu_{0} \Bigl(\sum_{|k|\neq 0} \omega^{2s}|k|^{2s} |b_{k}|^{2}\Bigr)^{1/2}+\mu_{1} |w|^{2}_{L^{2}(0,T)^{N}} \nonumber \\
&\leq \mu_{0} |w|^{2}_{\mathbb{H}^{s}_{m,T}}+\mu_{1} |w|^{2}_{L^{2}(0,T)^{N}} \label{berti}
\end{align}
where $\mu_{1}=T^{\frac{N-2}{2}}>0$.

Taking into account  (\ref{regv77}), (\ref{berti}) and (\ref{eqYY}), we deduce that
\begin{align}
&||v_{m}(\cdot,0)|^{\beta+1}|^{2}_{L^{2^{\sharp}_{s}}(0,T)^{N}}-\mu_{1} ||v_{m}(\cdot,0)|^{\beta+1}|^{2}_{L^{2}(0,T)^{N}} \\
& \leq \mu_{0} ||v_{m}(\cdot,0)|^{\beta+1}|^{2}_{\mathbb{H}^{s}_{m,T}} \nonumber \\
&\leq \mu_{0}  |||v_{m}|^{\beta+1}||_{\mathbb{X}^{s}_{m,T}}^{2} \nonumber \\
& \leq \mu_{0} \Bigl[ c_{\beta}(c+M)||v_{m}(\cdot,0)|^{\beta+1}|_{L^{2}(0,T)^{N}}^{2} \nonumber \\
&+c_{\beta}\varepsilon(M)||v_{m}(\cdot,0)|^{\beta+1}|_{L^{2^{\sharp}_{s}}(0,T)^{N}}^{2} \Bigr]  \label{iterbeta}.
\end{align}

Choosing $M$ large so that $c_{\beta} \mu_{0} \varepsilon(M)<\frac{1}{2}$, by (\ref{iterbeta}) we obtain
\begin{equation}\label{iter77}
||v_{m}(\cdot,0)|^{\beta+1}|_{L^{2^{\sharp}_{s}}(0,T)^{N}}^{2} \leq 2[\mu_{0}c_{\beta}(c+M)+\mu_{1}] ||v_{m}(\cdot,0)|^{\beta+1}|^{2}_{L^{2}(0,T)^{N}}.
\end{equation}
Let us notice that, by (\ref{baroni}) and $\mathbb{H}^{s}_{T}\subset L^{2^{\sharp}_{s}}(0,T)^{N}$, we get
\begin{align}\label{bessi}
|v_{m}(\cdot,0)|_{L^{2^{\sharp}_{s}}(0,T)^{N}} \leq K'''(\delta),
\end{align}
for any $m<m_{0}$.
By applying (\ref{iter77}) with $\beta+1=\frac{N}{N-2s}$ (that is $\beta=\frac{2s}{N-2s}$)  and by using (\ref{bessi}) we have that
$$
||v_{m}|^{\frac{N}{N-2s}}|_{L^{2^{\sharp}_{s}}(0,T)^{N}}^{2} \leq 2[c_{\frac{2s}{N-2s}} \mu_{0} (c+M)+\mu_{1}] K'''(\delta)^{\frac{2N}{N-2s}},
$$
and taking the limit as $m \rightarrow 0$, we deduce $v(\cdot,0)\in L^{\frac{2N^{2}}{(N-2s)^{2}}}(0,T)^{N}$.

By using the formula (\ref{iter77}), we find, after $k$ iterations, that $v(\cdot,0)\in L^{\frac{2N^{k}}{(N-2s)^{k}}}(0,T)^{N}$ for all $k \in \mathbb{N}$. Then $v(\cdot,0)\in L^{q}(0,T)^{N}$ for all $q\in[2,\infty)$, and by invoking Proposition $3.5.$ in \cite{FallFelli}, we can conclude that $v\in \mathcal{C}^{0,\alpha}([0,T]^{N})$, for some $\alpha \in (0,1)$.

redpajama_set_name: RedPajamaArXiv


text:
SOA Watch activists vow to keep their hands on the plow
No arrests with sparse crowd at SOA Watch protest
bordermobilization.jpg
SOA Watch border vigil
October 7, 2016 to October 10, 2016
SOA Watch is moving their vigil to the US/mexican border. In October we will gather on both sides of the wall and raise our voices for an end of the militarization of the Americas!
Currently we are still in the planning process and open for your ideas. If you are interested, would like to get involved or have any other questions email Sophie[at]irtfcleveland.org
RRN Letter: Mexico, 10/1/2016
Break-in and vandalism of Jesuit-sponsored Kino Migrant Center and death threats to staff in Nogales, Sonora (US border).
Killing on June 19 of journalist Elidio Ramos Zárate, age 45, a crime reporter for the regional daily newspaper El Sur, who was fatally shot in the neck by two unknown assailants driving a motorcycle.
Murder of journalist Elidio Ramos Zárate, a crime reporter for the regional daily newspaper EL Sur. Ramos Zárate was reporting on several blockades and buses set on fire by teachers protesting in Juchitán. Ramos was the third journalist killed in Oaxaca this year.
detention of Alvaro García, a leader of the Victim's Association of Magdalena Medio (ASORVIMM), and criminalization of other community leaders in Bella Unión and El Guayabo in the municipality of Puerto Wilches in Santander Department.
2016_06_26_afgj-telesur_mexico_letter_wo.pdf
Murder of journalist Francisco Pacheco Beltrán, from Taxco, Guerrero state. He was a reporter for several regional newspapers and a regular contributor to a radio station.
Disappearance of Fidencio Gómez Sántiz, an indigenous member of the National Front for Socialism (FNLS). He was last seen on March 5, accompanied by his wife and son at the Jet Ja transport terminal near the local market of Ocosingo, Chiapas State. He has not been seen since.
unjust imprisonment and maltreatment of Damián Gallardo Martinez, a teacher who has worked to defend the rights of indigenous communities and the right to education in the Mixe and Zapoteca regions of the Oaxaca Hills.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Extract file name from a file path starting with the given string I have a given path. For eg., it looks like C:\documents. I have extracted this path name to a cell in Excel. Say, I have 5 files inside it. I want to extract the name of the file starting with "S".
Is there a way to implement this without using VBA?
Can somebody help me, please... TIA.

redpajama_set_name: RedPajamaStackExchange


text:
import React from 'react';
import { shallow, render } from 'enzyme';
import MessageList from '../../components/protected/MessageList.jsx';

describe('MessageList components', () => {
  const group = { groupId: 'TUsUaucS5nb5kuOP6SpvTUcX9v3',
    groupName: 'Test Group' };
  const readUsers = [{ K345SHRKSDFKDSDGjhe5: 'Adeola' }, { K95745SHRKSDFKDSDGjhe5: 'Kola' }];
  const message = [{ messageBody: 'Test message',
    postedBy: 'testUser@gmail.com',
    postedByDisplayName: 'Test User',
    postedon: '',
    priority: 'Normal' }];
  const component = shallow(<MessageList selectedGroup={group} readUsers={readUsers} messages={message} />);
  it('should match snapshot test', () => {
    expect(component).toMatchSnapshot();
  });

  it('should render', () => {
    expect(component).toBeDefined();
  });
  it('Should contain three div', () => {
    expect(component.find('div').length).toEqual(3);
  });
  it('should recieve props', () => {
    expect(Object.keys(component.props()).length).toBeGreaterThan(0);
  });
});

redpajama_set_name: RedPajamaGithub


text:
package com.client.core.resttrigger.controller.clientcontact;

import com.bullhornsdk.data.model.entity.core.standard.ClientContact;
import com.client.core.base.model.relatedentity.ClientContactRelatedEntity;
import com.client.core.base.workflow.node.TriggerValidator;
import com.client.core.resttrigger.controller.AbstractRestTriggerController;
import com.client.core.resttrigger.model.api.RestTriggerRequest;
import com.client.core.resttrigger.model.api.RestTriggerResponse;
import com.client.core.resttrigger.model.helper.impl.ClientContactRestTriggerHelper;
import com.client.core.resttrigger.workflow.traversing.ClientContactRestTriggerTraverser;
import org.apache.log4j.Logger;
import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.http.MediaType;
import org.springframework.stereotype.Controller;
import org.springframework.web.bind.annotation.RequestBody;
import org.springframework.web.bind.annotation.RequestMapping;
import org.springframework.web.bind.annotation.RequestMethod;
import org.springframework.web.bind.annotation.ResponseBody;

import java.util.List;
import java.util.Map;
import java.util.Optional;

@Controller
@RequestMapping("/resttrigger/clientcontact/*")
public class ClientContactRestTriggerController extends AbstractRestTriggerController<ClientContact, ClientContactRestTriggerHelper, ClientContactRestTriggerTraverser> {

    private final Logger log = Logger.getLogger(ClientContactRestTriggerController.class);

    @Autowired
    public ClientContactRestTriggerController(Optional<List<TriggerValidator<ClientContact, ClientContactRestTriggerHelper, ClientContactRestTriggerTraverser>>> triggerValidators) {
        super(ClientContact.class, triggerValidators, ClientContactRelatedEntity.values());
    }

    /**
     * Called when client contact is added
     *
     * @param body
     *
     * @return the json parsed form response message
     */
    @RequestMapping(value = { "add" }, method = RequestMethod.POST, produces = { MediaType.APPLICATION_JSON_UTF8_VALUE })
    @ResponseBody
    public RestTriggerResponse addEntity(@RequestBody String body) {
        log.info("---------------------------- Starting Client Contact Add Validation Process From Rest Trigger ----------------------------------------");

	    Map<String, Object> valuesChanges = (Map<String, Object>) convertToMap(body).get("data");

	    RestTriggerRequest<ClientContact> restTriggerRequest = convertToObject(body);

	    Integer entityID = restTriggerRequest.getMeta().getEntityId();
	    Integer updatingUserID = restTriggerRequest.getMeta().getUserId();

	    ClientContactRestTriggerTraverser traverser = new ClientContactRestTriggerTraverser(entityID, valuesChanges, updatingUserID, false, getRelatedEntityFields());

	    return handleRequest(traverser, valuesChanges);
    }

    /**
     * Called when client contact is edited
     *
     * @param body
     *
     * @return the json parsed form response message
     */
    @RequestMapping(value = { "edit" }, method = RequestMethod.POST, produces = { MediaType.APPLICATION_JSON_UTF8_VALUE })
    @ResponseBody
    public RestTriggerResponse editEntity(@RequestBody String body) {
        log.info("---------------------------- Starting Client Contact Edit Validation Process From Rest Trigger ----------------------------------------");

        Map<String, Object> valuesChanges = (Map<String, Object>) convertToMap(body).get("data");

        RestTriggerRequest<ClientContact> restTriggerRequest = convertToObject(body);

	    Integer entityID = restTriggerRequest.getMeta().getEntityId();
        Integer updatingUserID = restTriggerRequest.getMeta().getUserId();

	    ClientContactRestTriggerTraverser traverser = new ClientContactRestTriggerTraverser(entityID, valuesChanges, updatingUserID, true, getRelatedEntityFields());

        return handleRequest(traverser, valuesChanges);
    }

}


I caught myself listening to the music of Meyniak, a Zimbabwean based Hip-hop artiste who is signed to Big Shots (G-Records). The music was powerful and resonated with m mind at a soulful level. I figured that Meyniak is more than the music.
His music is a paintbrush and our minds are the canvas. His debut album Before Dawn The Sire is probably one of the best album created in a while in Zimbabwe. It is a 2014 production but I had never talked about it because it was so deep I needed time to get acquainted with it. What I like is he didn't create an album to attract radio play and sacrifice the soul of the art. The artiste gave his time to create dope lyrics that spark the mind of a listener.
Bar for bar he has dethroned many that do get the recognition and when you create beautiful music like this it is not about the statistics or public image. Songs like Angels and Alive which both feature a talented musician both question and challenge popular belief or perceptions in an intellectual way. I think Meyniak is at home with these concepts.
Of course I smiled on hearing the mention on Crank "Mcpotar told me son start ba brand new order," but Angels is a personal favourite. Crank on the other hand is very lyrical and features Sharky who obviously ran his own 16. I don't like to compare the 2 but I'd say Meyniak to English bars is what Sharky is with Shona bars. The 2 artistes have worked together, developed each other and compliment each other. It is pretty sad when blogs and so forth (like this one) do not pick up on Meyniaks new material as fast as Sharky's but I guess then it has been a matter of accessibility.
Meyniak is incredible with delivery and syllables and he is an unsung hero in G-Records perhaps because of his busy schedule with Scholarly stuff he hasn't had much time to build ground on facebook on twitter. Talking about the art however he offers a lot of competition lyrically to celebrated artistes like Synik, Sharky, Tehn Diamond, Kapital K and Karma.
Talent alone however doesn't seem to create the stardom we need but given the right advice or agents he can indeed cash in on underground and mainstream success. With no doubt Before Dawn The Sire was better than most projects even nominated for Hip-hop awards but of course a lot is considered at awards and his album didn't match those criterion as far as mileage and visibility.
It makes one wish there were awards based on the art only minus the number of club hits, rotation and so forth from a particular album.
With all this said, support the artiste by coming through to this link to get his music.


you can use Android APIs while also supporting older versions by adding conditions to your code that check for the system API level before executing APIs not supported by your minSdkVersion.
Basically, API level means the Android version.
This defines for which version you are targeting your application and what is going to be the minimum level of android version in your application will run.
For setting Minimum level and Maximum level android studio provides two terminologies.
minSdkVersion means minimum Android OS version that will support your app and targetSdkVersion means the version for which you are actually developing your application.
Your app will be compatible with all the version of android which are falling between minimum level SDK and target SDK.


text:
Milichia prosaetes är en tvåvingeart som beskrevs av Collin 1922. Milichia prosaetes ingår i släktet Milichia och familjen sprickflugor.

Artens utbredningsområde är Nigeria. Inga underarter finns listade i Catalogue of Life.

Källor

Sprickflugor
prosaetes

redpajama_set_name: RedPajamaWikipedia


Are you a young water professional or PhD student working in the area of urban water management or urban planning? Then the Young Scientist Workshop 'Water Challenges of Urbanisation' during the International Water Week Amsterdam 2011 should be of interest to you!
The Young Scientist Workshop is an intensive scientific event students and international young professionals from the water industry and public organisations. The topic of the Workshop concerns the challenges of providing water systems in urbanizing areas, a cross-cutting topic covering water management, water supply, sanitation and spatial planning.
Apart from exchanging ideas with other professionals, you will work with colleagues from all over the world on a joint vision of the challenges and solutions to urbanisation problems. The outcome will be reported in a memorandum made available to all conference delegates, and published in a journal after the conference.
Before the workshop you are asked to prepare a scientific paper and/or case study addressing a specific water problem integral to the growth of cities under the stress of climate change. After review the selected papers will be published in the IWW-YSW proceedings. You and the other participants will present posters in the exhibition hall of the conference. The workshop is scientifically supervised by several professors involved in the water sector and some senior water professionals who chair the workshop sessions and the discussions.
You are invited to submit an abstract of about 500 words to the conference secretariat before 30 June 2011. The Young Scientist Workshop focus on urbanisation under pressures of growing population and climate change.
The best paper will be selected by a scientific jury and the winner will be announced during the IWW. The prize associated with the award is 1500 euro.
A scientific committee will review the abstracts. Notification of acceptance will be shortly after submission of the abstract and before 25 July. Full paper manuscripts are expected to be submitted before 1 September 2011.
More information can be found on the YSW website.


Paul Goodwin has collaborated with Monica de Miranda on Military Road, a video tracing the path of a road surrounding Lisbon. The remains of this 45km long road have now been occupied by makeshift homes built by recent immigrants. Thus, the area is considered the city's ghetto. Historically the army used the road to protect against English and French invaders – today it still acts as a fortress against 'invading foreigners', keeping immigrant populations on the margins.
Developed collaboratively with local communities, Military Road is a reminder of how cities function and continue to function in their engagement with immigrant populations. The work highlights the impact of globalisation in the creation of multi- directional migrations of people, cultures and ideas. Impacting on geographic and cultural transformations in the spatial organisation of the world and the city, from a place of localities into a space of fluxus and multiple movements of people.


text:
David Friedman Denies Calling For Ouster Of Abbas
March 29, 2018 By JTA
Image by Getty Images
JERUSALEM (JTA) — U.S. Ambassador to Israel David Friedman said reports about an interview in which he suggested that the White House was waiting for someone to replace Palestinian Authority President Mahmoud Abbas misrepresented his comments.
The interview will be published in Shevi'i, a Religious-Zionist newspaper that comes out on Fridays. Excerpts of the interview were released on Wednesday.
"If Abu Mazen is not interested in negotiating, I am sure that someone else will want to," Friedman said in the released excerpts. Abu Mazen is Abbas' nom de guerre. "If Abbas creates a vacuum, I am convinced that someone else will fill it, and then we will move forward" with the peace process.
On Thursday, Friedman clarified his comments in a tweet.
"I was misquoted in various reports stemming from an interview that published today. The United States is not seeking 'to replace' Mahmoud Abbas. It is for the Palestinian people to choose its leadership," the tweet said.
Abbas' five-year term as head of the Palestinian Authority ended in 2009; no new elections have been called and he remains in his post.
Abbas cut ties with Friedman and U.S. peace negotiators in December, when President Donald Trump recognized Jerusalem as Israel's capital and said he would move the U.S. embassy there.
Israeli-Palestinian Conflict


text:
Sunny Deol shares first look of Tara Singh from 'Gadar 2'
Tara Singh
23 December 2021, 11:47 am ·1-min read
Indian actor and politician
Sikh political and religious leader
Sunny Deol's first look as Tara Singh in 'Gadar 2' (Image source: Instagram)
New Delhi [India], December 23 (ANI): While announcing the first schedule wrap, actor Sunny Deol on Thursday shared his first look as Tara Singh from the upcoming film 'Gadar 2', co-starring Ameesha Patel in the lead role.
Taking to his Instagram handle, Sunny shared two pictures of himself as Tara Singh and wrote, "Only a fortunate few get to bring amazing characters back to life. Presenting Tara Singh 20 years later! Wrapped the first schedule of #Gadar2. Feeling blessed. #Gadar #TaraSingh."
'Gadar 2' is the sequel to the 2001 blockbuster hit romantic movie 'Gadar: Ek Prem Katha'. The first film mainly revolves around Tara Singh (portrayed by Sunny), a Sikh truck driver from Amritsar, who falls in love with Sakina (portrayed by Ameesha), a Muslim girl hailing from a political family in Lahore, Pakistan. Late Amrish Puri was also a part of the first installment.
'Gadar 2' will also star film director Anil Sharma's son Utkarsh Sharma in the lead role alongside Sunny and Ameesha. Utkarsh had played the role of Sunny and Ameesha's son in the first part.
The film went on floors in December in Himachal Pradesh. (ANI)
Reunion Dinner review: A CNY farce that hits home the meaning of family
Although it's a comedy, Reunion Dinner harks back to the purpose of the titular Chinese New Year custom.
Are sabbaticals one solution for the Great Resignation? This restaurant chain seems to think so
What benefits do restaurant owners need to offer in order to attract and retain personnel? Faced with a labor shortage, one American casual dining chain has come up with a solution that is both surprising and appealing: offering employees the possibility to take a sabbatical!
Now you can cycle sitting comfortably, protected from the elements thanks to a new ebike invention from Norway
The Norwegian startup Podbike is launching its first hybrid vehicle -- half bike, half car. Classed as an electric bike, it lets riders enjoy the benefits of cycling with a new level of comfort. Production is expected to start in spring 2022.
'Scream' reigns supreme, ousting 'Spider-Man' in N.America
New Paramount release "Scream" leapt to the top of the North American box office this weekend, showing the continuing lure of horror films while finally toppling "Spider-Man," industry watcher Exhibitor Relations reported Sunday.
Spotify still dominates the music streaming market, YouTube Music appeals to Gen Z
Spotify, Deezer, Apple Music, Amazon Music... Online listening platforms continue to grow, becoming the main means of music distribution. To such an extent that subscriptions to these services increased by 26.4% in the second quarter of 2021, according to a study by analytics firm Midia Research.


Ed and Betty Denman got into the motorcycle cargo trailer manufacturing in 1991 under the name of Country Trailers. Ed slowly made improvements to the design until he created the perfect cargo trailer. They finalized the design in 1998 and decided to have a "name that trailer" contest. The name that won was "The Breeze". And that's what their customers said; "it pulls like a breeze; you don't even know it's behind you". The Breeze became a very popular cargo trailer and many were pleased with their purchase.
In 2006, Ed made another improvement to the Breeze. All lights were changed to super bright LEDs.
In some instances, the lights on the trailer were more visible than the bike's lights.
Ed sold a Bright Blue Metallic Breeze to me in Spring 2007 to match my 2005 Goldwing. It has faithfully followed me across this country. It was such a good product that I ended up helping and working for Country Trailers at several rallies. Several times Ed asked me when I was going to buy the company from him so that he could retire. In 2013 Ed and Betty approached me and said it was time for them to make a change and would I be interested in purchasing the Breeze line. We worked out a deal where I will be producing the Breeze and carrying on the excellent service that Ed and Betty have given to their customers for the last 22 years.
We are pleased that the Denman's have entrusted me with their creation to move forward and continue offering the Breeze cargo trailer. We are excited about what the future holds for us!


text:
GVT presents Macbeth in HD
In partnership with The Met: Live in HD series, Greenbrier Valley Theatre (GVT) presents a live transmission of Giuseppe Verdi's Macbeth, Saturday, Oct. 11, at 12:55 p.m. The broadcast will run approximately 3 hours 15 minutes with one intermission. A complimentary lecture by Dr. James Caplinger from the University of Charleston will start at 12:15 p.m. For tickets, call the GVT Box Office at 304-645-3838 or visit www.gvtheatre.org.
The Met: Live in HD returns for the 2014-15 season with Adrian Noble's chilling production of Verdi's adaptation of Shakespearean tragedy Macbeth. Noble's production is the third staging in Met history.
Star soprano Anna Netrebko as the mad and murderous Lady Macbeth alongside Željko Lučić as the ill-fated Macbeth. The pair received rave reviews during their Met run, includ
ing this from NY Classical Review.
"This was Anna Netrebko's first Lady Macbeth at the Met, and she grabbed the role by the throat and did not let go." Željko Lučić gave a "dramatically compelling and vocally nuanced performance… His final, rage-fueled call to battle was harrowing. Luisi led a thrilling performance from the pit, weaving spooky atmospheres and wringing every drop of drama from the score."
Anna Netrebko as Lady Macbeth and Željko Lučić in title role of Verdi's Macbeth (Photo courtesy Marty Sohl/ Metropolitan Opera)
Literary Tea at GVT
Freefall Festival at Snowshoe


text:
Plan B group continues protest over holidays
A small group will continue their protest at the Plan B highway construction site in the Bonshaw area over Christmas.
CBC News · Posted: Dec 24, 2012 7:45 PM AT | Last Updated: December 26, 2012
Plan B protestor
Raw video of Plan B protestor Cindy Richards. 3:50
Cindy Richards and Janet MacLeod said they're watching for any potential environmental damage.
Richards and MacLeod have been camped out at the site in a teepee and in a trailer for weeks. Richards said they're making a difference.
"It's still a lot of public interest surrounding this. I think that we're making a difference," said Richards. "I think they're tightening up their mitigations and that things are being done at a better level that they would have, had we not been monitoring this."
Both Richards and MacLeod said they'll spend Christmas and the rest of winter on the site.
Province releases Plan B properties list


So Señor Kaos is getting official with it–he's prepping a soon to be released EP featuring the song, "Girls Rock Too" which has been featured on his last mixtape, Swagger Is Nothing, Talent Is Everything. The Girls Rock Too EP released by High Water Music, will feature the title track including instrumentals, remixes and clean versions of the cuts: "When I'm Gone" and "Automatic Classic".
"I made this song with hopes to enlighten people of the female accomplishments that have been forgotten about. You have adults with children who refuse to let their kids listen to the music because of the messages that are associated with it.
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