Asymptotic behavior of solutions to elliptic problems with Robin boundary conditions

Abstract

In this paper, we investigate the asymptotic behavior, as $β\to 0$, 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 $p \ge 0$, $β> 0$, and $Ω$ is a bounded smooth domain. We will prove that, for all $p\ge0$, the solution $u_β$ behaves like a constant as $β\to0$. However, the value of this constant is strongly influenced by the value of $p$. Indeed, \begin{itemize} \item if $0 \le p < 1$, $u_β$ blows up uniformly in $Ω$ as $β\to 0$. \item if $p=1$ (eigenvalue problem), $u_β$ converge to a constant. \item if $p>1$ $u_β$ 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 $Ω$ is a ball and $0<β<\frac{2}{p-1}$ we prove the existence of a radial positive solution.