具有臨界Sobolev－Hardy項(xiàng)的擬線性p－重調(diào)和方程解的存在性

任艷 桑彥彬

摘要：為了研究一類帶有Hardy項(xiàng)和多臨界Sobolev-Hardy指數(shù)的擬線性p-重調(diào)和方程解的存在性，借助于Ekeland變分原理，給出上述問題解的存在性定理。首先，將方程對(duì)應(yīng)的變分泛函定義在約束集Mη（通常稱為Nehari流形）上，使得該泛函下方有界。其次，利用纖維映射將上述集合Mη劃分為M+η，M0η和M-η等3部分，并分別研究每部分的性質(zhì)，證明了M+η和M-η中泛函極小值的存在性。最后，利用隱函數(shù)定理，得到在參數(shù)滿足一定條件下，存在極小化序列{un}，滿足（PS）c條件，從而完成了該方程解的存在性的證明。所得結(jié)論可為判定解的結(jié)構(gòu)和性質(zhì)提供理論依據(jù)。

關(guān)鍵詞：非線性泛函分析;臨界Sobolev-Hardy項(xiàng);擬線性p-重調(diào)和方程;Ekeland變分原理;解的存在性

中圖分類號(hào)：O175.25文獻(xiàn)標(biāo)志碼：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(jié)論

本文討論了一類具有臨界指數(shù)的p-重調(diào)和方程，運(yùn)用變分方法和Ekeland變分原理，建立了其解的存在性定理，可為判定解的結(jié)構(gòu)和性質(zhì)提供理論依據(jù)。

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