Ground Solutions Of Nonlinear Maxwell’s Equations In Asymmetric Media

In this dissertation,by discussing Maxwell’s equations with Kerr-like nonlinear terms in non-cylindrical symmetric media,a new semi-linear elliptic equation is derived where f(x,u)=(?)uF(x,u),Ω(?)R2.Using the Hilbert-Schmidt theory,the operator L:dom(L)(?)isgiven L2(Ω)6→L2(R2)6 The spectrum σ(L)={0,λ1,λ2,λ3,…,λn,…},where the eigenvalue 0 has infinite multiplicity And the corresponding eigenvectors belong to Each eigenvalue in W;{λn}n=1∞ has a finite multiplicity,and the corresponding eigenvector belongs to V,and meet 0<λ1<λ2<λ3 ≤…≤λn→∞.Since the kernel space of the operator L is infinite-dimensional,the semi-linear equation(1)The energy functional of is strongly indeterminate.We construct the appropriate Sobolev space X=V ⊕W,and use the variational method to give the equation(1)exists for the ground state solution.That is,when λ<0 orλ ≤0,assuming that the nonlinear term F satisfies certain conditions,the equation(1)has a ground state solution.Furthermore,if the nonlinear term F is even with respect to u,then J has an unbounded sequence of critical values.