Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain

Abstract

We find solutions $E:\Omega \to {\mathbb{R}}^3$ of the problem \[ \left\{\begin{aligned} &\nabla\times(\nabla\times E) + λE = \partial_E F(x,E) &&\quad \text{in}Ω\\ &ν\times E = 0 &&\quad \text{on}\partialΩ\end{aligned} \right. \] on a simply connected, smooth, bounded domain $\Omega \subset {\mathbb{R}}^3$ with connected boundary and exterior normal $\nu :\partial \Omega \to {\mathbb{R}}^3$. Here $\nabla \times$ denotes the curl operator in ${\mathbb{R}}^3$, the nonlinearity $F:\Omega \times {\mathbb{R}}^3\to \mathbb{R}$ is superquadratic and subcritical in $E$. The model nonlinearity is of the form $F(x,E)=\Gamma (x)|E{|}^p$ for $\Gamma \in L^{\infty }(\Omega )$ positive, some $2<p<6$. It need not be radial nor even in the $E$-variable. The problem comes from the time-harmonic Maxwell equations, the boundary conditions are those for $\Omega$ surrounded by a perfect conductor.

27 pages

Cited by in corpus (12)

• Ground states of a system of nonlinear Schrödinger equations with periodic potentials
• Multiple solutions to a nonlinear curl-curl problem in ${\mathbb{R}}^3$
• Existence and nonexistence of solutions for p(x)-curl systems arising in electromagnetism
• The Brezis-Nirenberg problem for the curl-curl operator
• Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities
• Cylindrically Symmetric Ground State Solutions for Curl-Curl Equations with Critical Exponent
• A Sobolev-type inequality for the curl operator and ground states for the curl-curl equation with critical Sobolev exponent
• Solutions of the fractional Schrödinger equation with sign-changing nonlinearity
• Multiple solutions to cylindrically symmetric curl-curl problems and related Schrödinger equations with singular potentials
• The fractional Schrödinger equation with Hardy-type potentials and sign-changing nonlinearities
• Generalized linking-type theorem with applications to strongly indefinite problems with sign-changing nonlinearities
• Travelling waves for Maxwell's equations in nonlinear and symmetric media