摘要
We survey recent results on ground and bound state solutions E:?→R^3 of the problem {▽(▽×E)+}λE=|E|^(P-2)E in Ω,v×E=0 on Ω on a bounded Lipschitz domain ??R^3,where?×denotes the curl operator in R^3.The equation describes the propagation of the time-harmonic electric field R{E(χ)e^(iwt)}in a nonlinear isotropic material ? withλ=-μεω~2≤0,where μ andεstand for the permeability and the linear part of the permittivity of the material.The nonlinear term|E|^(P-2)E with 2<p≤2*=6 comes from the nonlinear polarization and the boundary conditions are those for?surrounded by a perfect conductor.The problem has a variational structure;however the energy functional associated with the problem is strongly indefinite and does not satisfy the Palais-Smale condition.We show the underlying difficulties of the problem and enlist some open questions.
We survey recent results on ground and bound state solutions E： →R-3 of the problem {▽（▽×E）＋}λE=｜E｜-（P-2）E in Ω,v×E=0 on Ω on a bounded Lipschitz domain R-3,where ×denotes the curl operator in R-3.The equation describes the propagation of the time-harmonic electric field R{E（χ）e-（iwt）}in a nonlinear isotropic material ？ withλ=-μεω-2≤0,where μ andεstand for the permeability and the linear part of the permittivity of the material.The nonlinear term｜E｜-（P-2）E with 2〈p≤2＊=6 comes from the nonlinear polarization and the boundary conditions are those for？surrounded by a perfect conductor.The problem has a variational structure;however the energy functional associated with the problem is strongly indefinite and does not satisfy the Palais-Smale condition.We show the underlying difficulties of the problem and enlist some open questions.
基金
supported by the National Science Centre of Poland (Grant No. 2013/09/B/ST1/01963)
