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Multiple solutions for a nonhomogeneous Schrödinger-Poisson system with concave and convex nonlinearities. (English) Zbl 1465.35183

Summary: In this paper, we consider the following nonhomogeneous Schrödinger-Poisson equation \[ \begin{cases}- \Delta u +V(x)u+\phi(x)u =-k(x)|u|^{q-2}u+h(x)|u|^{p-2}u+g(x), & x\in \mathbb{R}^3,\\ -\Delta \phi =u^2, \quad \lim_{|x|\rightarrow +\infty}\phi(x)=0, & x\in \mathbb{R}^3, \end{cases} \] where \(1 < q < 2\), \(4 < p < 6\). Under some suitable assumptions on \(V (x), k(x), h(x)\) and \(g(x)\), the existence of multiple solutions is proved by using the Ekeland’s variational principle and the Mountain Pass Theorem in critical point theory.

MSC:

35J47 Second-order elliptic systems
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J10 Schrödinger operator, Schrödinger equation
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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