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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 $$\varepsilon$$.

##### MSC:
 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
##### Keywords:
asymptotic estimate; local mountain pass lemma
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##### References:
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