Standing waves with large frequency for 4-superlinear Schr\"odinger-Poisson systems

TL;DR

This paper studies standing wave solutions with large frequency for a 4-superlinear Schrödinger-Poisson system, overcoming challenges posed by indefinite potentials and weaker nonlinear conditions.

Contribution

It establishes existence of solutions for large frequencies under weaker nonlinear assumptions using variational methods like local linking and fountain theorems.

Findings

Existence of nontrivial solutions for large 

Unbounded sequence of solutions in the odd nonlinearity case

Solutions obtained despite indefinite potential and weaker nonlinear conditions

Abstract

We consider standing waves with frequency $\omega$ for 4-superlinear Schr\"odinger-Poisson system. For large $\omega$ the problem reduces to a system of elliptic equations in $\mathsf{R}^3$ 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.