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A singularly perturbed linear eigenvalue problem in $$C^1$$ domains. (English) Zbl 1061.35061
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 $$\gamma\to\infty$$.

##### 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
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