Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum
Let $\Omega\subset\mathbb{R}^{N}$ ($N\geq1$) be a smooth bounded domain, $a\in C(\bar{\Omega})$ a sign-changing function, and $0\leq q<1$. We investigate the Robin problem \[ \begin{cases} -\Delta u=a(x)u^{q} & \mbox{in $\Omega$},\\ u\geq0 & \mbox{in $\Omega$},\\ \partial_{\nu}u=\alpha u & \mbox{on $\partial \Omega$}, \end{cases} \] where $\alpha\in\lbrack-\infty,\infty)$ and $\nu$ is the unit outward normal to $\partial\Omega$. Due to the lack of strong maximum principle structure, this problem may have \textit{dead core} solutions. However, for a large class of weights $a$ we recover a \textit{positivity} property when $q$ is close to $1$, which considerably simplifies the structure of the solution set. Such property, combined with a bifurcation analysis and a suitable change of variables, enables us to show the following exactness result for these values of $q$: $(P_{\alpha})$ has \textit{exactly} one nontrivial solution for $\alpha\leq0$, \textit{exactly} two nontrivial solutions for $\alpha>0$ small, and \textit{no} such solution for $\alpha>0$ large. Assuming some further conditions on $a$, we show that these solutions lie on a subcontinuum. These results rely partially on (and extend) our previous work \cite{KRQU16}, where the cases $\alpha=-\infty$ (Dirichlet) and $\alpha=0$ (Neumann) have been considered. We also obtain some results for arbitrary $q\in\left[ 0,1\right) $. Our approach combines mainly bifurcation techniques, the sub-supersolutions method, and \textit{a priori} lower and upper bounds.