A singularly perturbed linear eigenvalue problem in \(C^1\) domains. (Q1880134)

For any \(\gamma>0\), set \[ \Lambda(\gamma)=\sup_{u\in H^1(\Omega)\setminus\{0\}}\frac{\gamma\int_{\partial\Omega}u^2-\int_\Omega| \nabla u| ^2}{\int_\Omega u^2}, \] where \(\Omega\) is a bounded domain in \(\mathbb R^n\) with boundary \(\partial\Omega\in C^1\). The supremum is attained by some positive function \(u_\gamma\in H^1(\Omega)\) , which is a weak solution of \[ \Delta u=\Lambda(\gamma)u\quad \text{in } \Omega,\qquad \frac{\partial u}{\partial\nu}=\gamma u\quad \text{on }\partial\Omega, \] where \(\nu\) is the outward unit normal vector on \(\partial\Omega\). The goal of this paper is to understand the asymptotic behavior of \(\Lambda(\gamma)\) as \(\gamma\to\infty\). Since \(\Lambda(\gamma)\to\infty\) when \(\gamma\to\infty\), this problem can be viewed as a singularly perturbed linear eigenvalue problem. The following theorems are proved. Theorem 1. \[ \lim_{\gamma\to\infty}\frac{\Lambda(\gamma)}{\gamma^2}=1 \] holds for any bounded \(C^1\) domain. Theorem 2. If \(a>1\), then \[ \Delta u=au \quad \text{in } \mathbb R_+^n,\qquad \frac{\partial u}{\partial x_n}=-u\quad \text{on }\partial \mathbb R_+^n, \] has no bounded nontrivial solution. Here \(a\) is the limit of \(\frac{\Lambda(\gamma)}{\gamma^2}\) (subject to a subsequence) as \(\gamma\to\infty\).