Some Studies Of Variational Perturbations For Gross-pitaevskii Equation With Periodic Potentials

In this paper,we study the following Gross-Pitaevskii equation with potential where ??{1.-1}.E?R ?,(x)?C0?(RN).and it is the real function.V(x)?C?(RN)is a periodic real function.We prove the following of the above Gross-Pitaevskii equation for some ranges of the parameters ?,and N:(i)the energy is not.affected by perturbation at infinity,that is,there exists a constant c such that where Vn is the cube of size n inRN,?wVn(u)is the local energy functional of perturbed equation-?u(x)+V(x)u(x)+ w(x)u(x)+?(x)|2u(x)=Eu(x),define(ii)the equation solutions are also not affected by perturbation at infinity,that is,u(x)satisfies-?u(x)+ V(x)u(x)w(x)u(x)+ ?|u(x)|2u(x)= Eu(x),then there exists a unperturbed periodic solution u0(x)such that where u0(x)satisfies-?u(x)+ V(x)u(x)+?|u(x)|2u(x)= Eu(x).