具有奇异非线性项的(p,q)-Laplace方程组的多解性  

Multiplicity of Solutions to a(p,q)-Laplace System with Singular Nonlinearities

作　　者：桑彦彬 贺露萱 SANG Yanbin;HE Luxuan(School of Mathematics,North University of China,Taiyuan 030051,China)

机构地区：[1]中北大学数学学院,太原030051

出　　处：《应用数学学报》2023年第6期845-864,共20页Acta Mathematicae Applicatae Sinica

基　　金：山西省基础研究计划资助项目(批准号:202103021224198);山西省高等学校青年科研人员培育计划资助项目。

摘　　要：该文将研究以下具有变号系数和奇异非线性项的(p,q)-Laplace方程组{-△pu-△qu=g(x)u^(-γ)+2α/α+βh(x)u^(α-1)v^(β),x∈Ω,-△pv-△qv=f(x)v^(-γ)+2β/α+βh(x)u^(α)v^(β-1),x∈Ω,u,v>0,x∈Ω,u=v=0,x∈■Ω,其中γ∈(0,1),α,β>1,1<q<p<α+β<p*=Np/N-p,f,g∈p*/Lp*+γ-1(Ω)均为非负函数,h∈/Lp*/p*-α-β(Ω)且{x∈Ω:h(x)>0}具有正测度.借助于Ekeland变分原理和一些分析技巧,建立了上述问题的多重解的存在性定理.当加权函数满足一定的限制条件时,获得了基态解的存在性.In recent years,much attention has paid to(p,q)-Laplace equation and system involving positive exponents and concave-convex nonlinearities.In this paper,we study a class of(p,q)-Laplace systems with singular nonlinearities.The main feature of our problem is to include negative exponents and sign-changing potential.Because the energy functional corresponding to above problem is not differentiable,the traditional variational tools can not be used to solve above problem,directly.Through Ekeland variational principle,Nehari manifold decomposition and some real analysis methods,the multiplicity theorem of weak solutions of above problem is established.Converge properties of minimizing sequence are discussed by Vitali converge Theorem and Fatau Lemma.Furthermore,when weighted functions satisfy certain conditions,the existence of ground state solution is also obtained.Owing to the presence of coupled terms,some delicate analysis will be needed to overcome singularity derived from negative exponents.Some uniform estimation of weighted coefficient functions will be given.Our works extend and improve results of the single equation of(p,q)-Laplace equation to coupled system.

关 键 词：(p q)-Laplace方程组 EKELAND变分原理 负指数 变号系数 

分 类 号：O175.2[理学—数学] O176.3[理学—基础数学]