Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8896752 | Journal of Functional Analysis | 2018 | 34 Pages |
Abstract
We look for solutions E:ΩâR3 of the problem{âÃ(âÃE)+λE=|E|pâ2Ein ΩνÃE=0on âΩ on a bounded Lipschitz domain ΩâR3, where âà denotes the curl operator in R3. The equation describes the propagation of the time-harmonic electric field â{E(x)eiÏt} in a nonlinear isotropic material Ω with λ=âμεÏ2â¤0, where μ and ε stand for the permeability and the linear part of the permittivity of the material. The nonlinear term |E|pâ2E with p>2 is responsible for the nonlinear polarisation of Ω and the boundary conditions are those for Ω surrounded by a perfect conductor. The problem has a variational structure and we deal with the critical value p, for instance, in convex domains Ω or in domains with C1,1 boundary, p=6=2â 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 λâ¤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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
JarosÅaw Mederski,