Radial and nonradial solutions of a strongly indefinite elliptic system on $\mathbb{R}^N$

TL;DR

This paper proves the existence of infinitely many radial and nonradial solutions for a strongly indefinite elliptic system on ^N, using symmetry principles and a generalized variational approach.

Contribution

It introduces a novel application of the Principle of Symmetric Criticality and a generalized Fountain Theorem to find multiple solutions of indefinite elliptic systems.

Findings

Existence of infinitely many solutions including nonradial ones.

Solutions are obtained under symmetry and growth conditions on the nonlinearity.

The method extends variational techniques to strongly indefinite problems.

Abstract

This paper is concerned with the following system of elliptic equations {equation*} \{{array}{ll} -\Delta u+u= F_u(|x|,u,v), & \hbox{} -\Delta v+v=- F_v(|x|,u,v), & \hbox{} \,\,\,\,\,u,v\in H^1(\mathbb{R}^N). & \hbox{} {array}. {equation*} It is shown that if $F$ is odd in $(u,v)$ and satisfy some growth conditions, then $(\mathcal{S})$ has infinitely many both radial and nonradial solutions. The proof relies on the Principle of Symmetric Criticality and a generalized Fountain Theorem for strongly indefinite functionals.