On the Hénon equation with a Neumann boundary condition: asymptotic profile of ground states
This paper is concerned with the asymptotic behaviour of ground state solutions to a Neumann boundary value problem for the Hénon equation \[ \begin{cases} - \Delta u + u = |x|^\alpha u^p&\text{in}\quad\Omega,\\ \frac{\partial u}{\partial \nu} = 0&\text{on}\quad\partial\Omega,\end{cases}\tag{$*$} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N\ge 2\), with \(C^2\)-smooth boundary \(\partial\Omega\) and \(\nu\) is the outward unit normal on \(\partial\Omega\). Assuming that \(\Omega\) lies entirely in the unit ball \(B(0,1)\) and \(\partial^* \Omega = \partial\Omega \cap \partial B(0,1) \ne \emptyset\), the authors show that the ground state solutions to (\(*\)) have different limiting behaviour as \(\alpha\to\infty\) depending on the interplay between the geometric properties of \(\partial^* \Omega\) and the growth rate \(p\in (1,2^* - 1)\), where \(2^*\) is the critical Sobolev exponent of the \(H^1(\Omega)\)-embedding. More precisely, the localization phenomenon of ground state solutions occurs for large \(\alpha\) in the following two situations: (i) if \(p\in (2_*-1,2^* - 1)\) and \(\partial^* \Omega\) is arbitrary, or (ii) if \(p\in (1,2_*-1]\) and \(\partial^* \Omega\) has measure zero, where \(2_*\) is the critical exponent of the Sobolev trace embedding.
scientific article; zbMATH DE number 6863567
scientific article