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On ground states for the Schrödinger-Poisson system with periodic potentials. (English) Zbl 1371.35041

Summary: This paper is concerned with the following Schrödinger-Poisson system \[ \begin{cases} -\Delta u+V(x)u-K(x)\phi(x)u= q(x)|u|^{p-2}u,&\text{in }\mathbb R^3,\\ -\Delta\phi=K(x)u^2,&\text{in }\mathbb R^3, \end{cases} \] where \(p\in(2,6)\), \(V(x)\in C(\mathbb R^3,\mathbb R)\) is a general periodic function, \(K(x)\) and \(q(x)\) are non-periodic functions. Under suitable assumptions, we prove the existence of ground state solutions via variational methods for strongly indefinite problems

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J10 Schrödinger operator, Schrödinger equation
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