具有临界Sobolev－Hardy项的拟线性p－重调和方程解的存在性

任艳 桑彦彬

摘要：为了研究一类带有Hardy项和多临界Sobolev-Hardy指数的拟线性p-重调和方程解的存在性，借助于Ekeland变分原理，给出上述问题解的存在性定理。首先，将方程对应的变分泛函定义在约束集Mη（通常称为Nehari流形）上，使得该泛函下方有界。其次，利用纤维映射将上述集合Mη划分为M+η，M0η和M-η等3部分，并分别研究每部分的性质，证明了M+η和M-η中泛函极小值的存在性。最后，利用隐函数定理，得到在参数满足一定条件下，存在极小化序列{un}，满足（PS）c条件，从而完成了该方程解的存在性的证明。所得结论可为判定解的结构和性质提供理论依据。

关键词：非线性泛函分析;临界Sobolev-Hardy项;拟线性p-重调和方程;Ekeland变分原理;解的存在性

中图分类号：O175.25文献标志码：A

Abstract：In order to study a class of quasilinear p-biharmonic equations with Hardy terms and multi-critical Sobolev-Hardy exponents， the existence theorem of the solutions to the above problem is established by means of the Ekeland variational principle. Firstly， to guarantee the variational functional is bounded from below， it is restricted on a set  Mη （usually called Nehari manifold）. Secondly， the set Mη  is divided into three parts  M+η， M0η  and M-η  by using fibering maps. Moreover， the existence of minimum in  M+η and M-η  is proved by studying the properties of the two subsets. Finally， by using implicit function theorem， it is found that there exists a minimizing sequence {un}  making the （PS）c  conditions hold when the parameters satisfy certain conditions. Therefore， the existence of the solutions to the problem is proved. The conclusions provide a theoretical basis for judging the structure and properties of the solutions.

Keywords：nonlinear functional analysis; critical Sobolev-Hardy terms; quasilinear p-biharmonic equations; Ekeland's variational principle; existence of the solution

3結论

本文讨论了一类具有临界指数的p-重调和方程，运用变分方法和Ekeland变分原理，建立了其解的存在性定理，可为判定解的结构和性质提供理论依据。

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