Nonlinear time-harmonic Maxwell equations in a bounded domain: Lack of compactness

We survey recent results on ground and bound state solutions E : O. R3 of the problem 8< :. x (. x E) + E = | E | p in O; x E = 0 on @ O on bounded Lipschitz domain O. R3, where. x denotes the curl operator in R3. The equation describes the propagation of the time-harmonic electric field R {E (x) e i!t} in nonlinear isotropic material O with = -"!2 6 0, where nd " stand for the permeabilitynd the linear part of the permittivity of the material. The nonlinear term | E | p with 2 < p 6 2 = 6 comes from the nonlinear polarizationnd the boundary conditionsre those for O surrounded by perfect conductor. The problem has variational structure; however the energy functionalssociated with the problem is strongly indefinitend does not satisfy the Palais-Smale condition. We show the underlying difficulties of the problemnd enlist some open questions.