Asymptotic behavior of solutions to elliptic problems with Robin boundary conditions

In this paper, we investigate the asymptotic behavior, as $\beta \to 0$, of positive solutions to the semilinear elliptic Robin problem \begin{equation*} \begin{cases} -Δu = u^p, & $\text{in }$ Ω,\ u > 0, & $\text{in }$ Ω,\ \frac{\partial u}{\partial ν} + βu = 0, & $\text{on }$ \partial Ω, \end{cases} \end{equation*} where $p\ge 0$, $\beta >0$, and $\mathrm{\Omega }$ is a bounded smooth domain. We will prove that, for all $p\ge 0$, the solution $u_{\beta }$ behaves like a constant as $\beta \to 0$. However, the value of this constant is strongly influenced by the value of $p$. Indeed, \begin{itemize} \item if $0\le p<1$, $u_{\beta }$ blows up uniformly in $\mathrm{\Omega }$ as $\beta \to 0$. \item if $p=1$ (eigenvalue problem), $u_{\beta }$ converge to a constant. \item if $p>1$ $u_{\beta }$ converge uniformly to zero. \end{itemize} In the critical and supercritical regime $p\ge \frac{N+2}{N-2}$, the existence of solutions is no longer guaranteed a priori. In this case, when $\mathrm{\Omega }$ is a ball and $0<\beta <\frac{2}{p-1}$ we prove the existence of a radial positive solution.