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} -\Delta u = u^p, & \text{in } \Omega,\\ u > 0, & \text{in } \Omega,\\ \frac{\partial u}{\partial \nu} + \beta u = 0, & \text{on } \partial \Omega, \end{cases} \end{equation*} where $p \ge 0$, $\beta > 0$, and $\Omega$ is a bounded smooth domain. We will prove that, for all $p\ge0$, the solution $u_\beta$ behaves like a constant as $\beta\to0$. 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 $\Omega$ 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 $\Omega$ is a ball and $0<\beta<\frac{2}{p-1}$ we prove the existence of a radial positive solution.