Seminarium KMMF "Teoria Dwoistości"

The nonlinear curl-curl problems have recently arisen in the search for exact propagation of electromagnetic waves in non-linear media modelled with Maxwell equations. The quintic effect leads to the following critical curl-curl problem:∇ × (∇ × u) = |u|^4 u,where u : R^3 → R^3 is the profile of the time-harmonic electric field. Ground states solutions of the problem are related with the optimizers of a Sobolev- type inequality involving the curl operator in R^3. We show that there is a ground state solution and infinitely many bound state solutions. Some symmetric properties of the problem and extensions to the p-curl-curl equation in the critical case with applications to zero modes of the Dirac equation will be also discussed.