Linear and semilinear eigenvalue problems in exterior domains (Q1567537)

Linear and nonlinear eigenvalue problems in exterior domains are studied. Namely, the eigenvalue problem \(-\Delta u=\lambda f(x)\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\), \(u(x)\to 0\) as \(|x|\to+\infty\), where \(\Omega\) is a exterior smooth domain in \(\mathbb{R}^n\) \((n\geq 3)\), \(0\in \mathbb{R}^n-\Omega\) and \(f>0\) is also smooth, is considered. Moreover, it is assumed that there exists \(P(|x|)\) positive such that \(f(x)\leq P(|x|)\) and \(\int_\Omega|x|^{2- n}P(|x|) dx< +\infty\). Then there is a unique eigenvalue \(\lambda_1>0\) which is simple and has an eigenfunction \(h(x)> 0\) which is minimal in the sense that \(\sup_x |h(x)|(1+|x|)^{n- 2}< +\infty\) and \(\lim_{|x|\to +\infty} h(x)(1+|x|)^{n- 2}= C\) for some \(C>0\). Since \(\Omega\) is unbounded, the proof becomes much more involved than usual. In particular, it is more difficult to show that the solution operator is compact; moreover, the Krein-Rutman theorem cannot be applied directly, another difficulty which should be circumvented. The nonlinear eigenvalue problem obtained by perturbating the right-hand side by \(\mu f(x)u^\gamma\), with \(0<\gamma< 1\) is also studied. If \(0\leq\lambda\leq \lambda_1\) and \(P\) satisfies now the condition \(\int_\Omega|x|^{\gamma(2- n)}P(|x|) dx<+\infty\), there is a positive minimal solution \(u\) such that \((1+|x|)^{n- 2}u(x)\to C\) as \(|x|\to +\infty\) for some \(C>0\). This is proved by using a sub-supersolution argument.