Local mountain passes for semilinear elliptic problems in unbounded domains
The authors consider the semilinear elliptic problem \[ \varepsilon^2 \Delta u- V(x) u+ f(u)= 0\quad\text{ in } \;\Omega,\quad u= 0\quad\text{on} \quad \partial\Omega,\quad u> 0,\tag{\(*\)} \] where \(\Omega\subset \mathbb R^n\) is a possibly unbounded domain. The function \(f\) is assumed to be of subcritical growth and \(f(\xi)/\xi\) is nondecreasing. The potential \(V\) is strictly positive and locally Hölder continuous. The main result of this paper states that there exists a positive solution of \((*)\) for sufficiently small \(\varepsilon> 0\), if \[ \inf_{G} V< \min_{\partial G} V \] holds for some domain \(G\) compactly contained in \(\Omega\). An asymptotic estimate for the solution is given, too. The proof of this result relies on a local mountain pass lemma. Since the energy functional associated to \((*)\) does not satisfy the usual Palais-Smale condition, the authors introduce a truncated functional, the critical points of which are also solutions of \((*)\) for small~\(\varepsilon\).
scientific article; zbMATH DE number 852856