Dual Variational Methods for a nonlinear Helmholtz system

TL;DR

This paper develops dual variational methods to prove the existence of nontrivial solutions for a coupled nonlinear Helmholtz system on ^N, providing conditions for the nontriviality of both solution components.

Contribution

It introduces a novel dual variational approach to establish solutions and characterizes parameter conditions for nontrivial solutions in a coupled Helmholtz system.

Findings

Existence of solutions in W^{2,p}(^N) established.

Necessary and sufficient conditions for nontrivial solutions derived.

Method applicable to nonlinear Helmholtz systems with coupling.

Abstract

This paper considers a pair of coupled nonlinear Helmholtz equations \begin{align*} -\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*} -\Delta v - \nu v = a(x) \left( |v|^\frac{p}{2} + b(x) |u|^\frac{p}{2} \right)|v|^{\frac{p}{2} - 2}v \end{align*} on $\mathbb{R}^N$ where $\frac{2(N+1)}{N-1} < p < 2^\ast$. The existence of nontrivial strong solutions in $W^{2, p}(\mathbb{R}^N)$ is established using dual variational methods. The focus lies on necessary and sufficient conditions on the parameters deciding whether or not both components of such solutions are nontrivial.