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 κ∈{1.-1}.E∈R ω,(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 κ,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,φwVn(u)is the local energy functional of perturbed equation-△u(x)+V(x)u(x)+ w(x)u(x)+κ(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)+ κ|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)+κ|u(x)|2u(x)= Eu(x).