Local mountain passes for semilinear elliptic problems in unbounded domains. (English) Zbl 0844.35032

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 \(\varepsilon\).


35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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