Standing waves with large frequency for 4-superlinear Schrödinger–Poisson systems

Abstract

We consider standing waves with frequency $$\omega $$ for 4-superlinear Schrödinger–Poisson system. For large $$\omega $$ , the problem reduces to a system of elliptic equations in $$\mathbb R ^3$$ with potential indefinite in sign. The variational functional does not satisfy the mountain pass geometry. The nonlinearity considered here satisfies a condition which is much weaker than the classical (AR) condition and the condition (Je) of Jeanjean. We obtain nontrivial solution and, in case of odd nonlinearity, an unbounded sequence of solutions via the local linking theorem and the fountain theorem, respectively.