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A positive solution of asymptotically periodic Schrödinger equations with local superlinear nonlinearities. (English) Zbl 1463.35245

Summary: In this paper, we investigate the following Schrödinger equation \[-\Delta u+V(x)u=\lambda f(u) \quad {\text{ in}} \ \mathbb{R}^N,\] where \(N\geq 3\), \(\lambda>0\), \(V\) is an asymptotically periodic potential and the nonlinearity term \(f(u)\) is only locally defined for \(\vert u\vert \) small and satisfies some mild conditions. By using Nehari manifold and Moser iteration, we obtain the existence of positive solutions for the equation with sufficiently large \(\lambda\).

MSC:

35J50 Variational methods for elliptic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B09 Positive solutions to PDEs
35D30 Weak solutions to PDEs
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