Dual Variational Methods for a nonlinear Helmholtz system

Abstract

This paper considers a pair of coupled nonlinear Helmholtz equations \begin{align*} -Δu - μu = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*} -Δv - νv = 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.

Published version. Contains minor revisions: Quote added, explanations on p.12 concerning F_{μν} = \infty, correction of exponent on p.18

References in corpus (1)

• Oscillating solutions for nonlinear Helmholtz Equations

Cited by in corpus (2)

• On a fourth order nonlinear Helmholtz equation
• Variational methods for breather solutions of Nonlinear Wave Equations