Lou, Yuan; Zhu, Meijun A singularly perturbed linear eigenvalue problem in \(C^1\) domains. (English) Zbl 1061.35061 Pac. J. Math. 214, No. 2, 323-334 (2004). 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 \(\gamma\to\infty\). Reviewer: Andrey Ivanovic Sedov (Magnitogorsk) Cited in 18 Documents MSC: 35P15 Estimates of eigenvalues in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B25 Singular perturbations in context of PDEs 35J25 Boundary value problems for second-order elliptic equations Keywords:bounded domain; weak solution; asymptotic behavior PDFBibTeX XMLCite \textit{Y. Lou} and \textit{M. Zhu}, Pac. J. Math. 214, No. 2, 323--334 (2004; Zbl 1061.35061) Full Text: DOI