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Ground states and multiple solutions for Choquard-Pekar equations with indefinite potential and general nonlinearity. (English) Zbl 1465.35248

Summary: This paper focuses on the study of ground states and multiple solutions for the following non-autonomous Choquard-Pekar equation: \[ \begin{cases} - \Delta u + V ( x ) u = ( W \ast F ( x , u ) ) f ( x , u ) , \quad x \in \mathbb{R}^N \quad( N \geq 2 ), \\ u \in H^1 ( \mathbb{R}^N ) , \end{cases}\] where \(V \in \mathcal{C}( \mathbb{R}^N, \mathbb{R})\). We consider first the case \(V\) changes sign which turns the problem into a indefinite case, and obtain the existence of nontrivial solution and infinitely many distinct pairs of solutions under a local super-linear condition assumed on the nonlinearity. For the case \(V\) is 1-periodic and positive, ground state solution and infinitely many solutions are established further by using the generalized Nehari manifold method. We finally give some non-existence criteria via a generalized Pohožaev identity established for the general potentials \(V\) and \(W\).

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
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