一类半线性退化Schr?dinger方程解的存在性

Existence of Solutions for a Class of Semi-linear Degenerate Schr?dinger Equation

椭圆型偏微分方程是一类非常重要的偏微分方程,目前对于半线性椭圆型偏微分方程已经有了相当丰富的研究,但是对于带有退化算子Δ_(γ)的椭圆型偏微分方程的研究不是很完善,还有很多值得研究的问题。该文主要研究如下半线性退化Schr dinger方程:-Δ_(γ)u+u=b(x)u p-1 u,x∈R N,u∈S 2γ(R N),其中1<p<2_(γ)^(*)-1,N 2,Δγ是退化椭圆算子。b(x)满足b(x)∈L∞(R N)且对任意的x∈R N有b(x)1,lim|x|→+∞b(x)=b∞=1。利用变分法以及山路定理证明半线性退化Schr dinger方程非平凡解的存在性。

As elliptic partial differential equation is a partial differential equation of great importance,there are quite a lot of researches focused on semi-linear elliptic partial differential equation at present.However,the research on elliptic partial differential equation with degenerate operatorΔγis far from being complete,and there are many problems worthy of study.This paper mainly talks about the semi-linear degenerate Schr dinger equation:-Δγu+u=b(x)|u|p-1 u,x∈R N,u∈S 2γ(R N),in the equation,1<p<2_(γ)^(*)-1,N 2;Δγis a degenerate elliptic operator;b(x)satisfies b(x)∈L∞(R N)and for every x∈R N,b(x)1,lim|x|→+∞b(x)=b∞=1.The existence of nontrivial solution for semi-linear Schr dinger equation is proved by the variational method and mountain pass theorem.