一类Schr?dinger-Poisson方程的约束极小元

A class of constrained minimal elements for Schr?dinger-Poisson equations

运用变分法研究一类Schrodinger-Poisson方程在指定L^2范数下极小元的存在性和不存在性.首先,利用Gagliardo-Nirenberg和Hardy-Littewood-Sobolev不等式并且选取试验函数做一些估计;其次,在对非线性项部分指标p的分类讨论中,通过极小化序列方法、紧嵌入引理、Ekeland变分原理、消失引理以及Pohozaev恒等式证明了约束极小元的存在性和不存在性.

The existence and the nonexistence of minimal elements with prescribed L^2-norm for a class of Schrodinger-Poisson equations are considered by using variational methods.Firstly,by using Gagliardo-Nirenberg and Hardy-Littewood-Sobolev inequalities and selecting testing functions,some estimates are obtained.Secondly,in the discussion on the classification of exponent p of nonlinearity,by using the method of minimizing sequence,compact embedding lemma,Ekeland's variational principle,vanishing lemma,Pohozaev's identity,the existence and the nonexistence of constrained minimal elements are proved.