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Existence and multiplicity results for the nonlinear Schrödinger-Poisson systems. (English) Zbl 1241.35192

Summary: We study the existence and multiplicity results for the nonlinear Schrödinger-Poisson systems

-Δu+V(x)u+K(x)ϕ(x)u=f(x,u),in 3 ,-Δϕ=K(x)u 3 ,in 3 ·(*)

Under certain assumptions on V,K and f, we obtain at least one nontrivial solution for (*) without assuming the Ambrosetti and Rabinowitz condition by using the mountain pass theorem, and obtain infinitely many high energy solutions when f(x,·) is odd by using the fountain theorem.

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
35A01Existence problems for PDE: global existence, local existence, non-existence
35A15Variational methods (PDE)
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