The Brezis-Nirenberg problem for the curl-curl operator

Abstract

We look for solutions $E:\Omega \to {\mathbb{R}}^3$ of the problem $$\{\begin{array}{rlrlrlr}& \nabla \times (\nabla \times E)+\lambda E=|E{|}^{p-2}E& & \text{in }\Omega & \nu \times E=0& & \text{on }\partial \Omega \end{array}$$ on a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^3$, where $\nabla \times$ denotes the curl operator in ${\mathbb{R}}^3$. The equation describes the propagation of the time-harmonic electric field $\Re \{E(x)e^{i\omega t}\}$ in a nonlinear isotropic material $\Omega$ with $\lambda =-\mu \epsilon {\omega }^2\le 0$, where $\mu$ and $\epsilon$ stand for the permeability and the linear part of the permittivity of the material. The nonlinear term $|E{|}^{p-2}E$ with $p>2$ is responsible for the nonlinear polarisation of $\Omega$ and the boundary conditions are those for $\Omega$ surrounded by a perfect conductor. The problem has a variational structure and we deal with the critical values $p$, for instance, in convex domains $\Omega$ or in domains with ${\mathcal{C}}^{1,1}$ boundary $p=6=2^{\ast }$ is the Sobolev critical exponent and we get the quintic nonlinearity in the equation. We show that there exist a cylindrically symmetric ground state solution and a finite number of cylindrically symmetric bound states depending on $\lambda \le 0$. We develop a new critical point theory which allows to solve the problem, and which enables us to treat more general anisotropic media as well as other variational problems.

to appear in J. Funct. Anal

References in corpus (2)

• Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain
• Ground states of a system of nonlinear Schrödinger equations with periodic potentials

Cited by in corpus (5)

• Multiple solutions to a nonlinear curl-curl problem in ${\mathbb{R}}^3$
• A Sobolev-type inequality for the curl operator and ground states for the curl-curl equation with critical Sobolev exponent
• Multiple solutions to cylindrically symmetric curl-curl problems and related Schrödinger equations with singular potentials
• Nonlinear curl-curl problems in ${\mathbb{R}}^3$
• Nonlinear time-harmonic Maxwell equations in domains