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    188宝金博页面版: Existence of Weak Solutions to a Class of Singular Elliptic Equations

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    内容提示： Mediterr. J. Math. 13 (2016), 4917–4927DOI 10.1007/s00009-016-0782-91660-5446/16/064917-11published onlineAugust 16, 2016c ? Springer International Publishing 2016Existence of Weak Solutions to a Class ofSingular Elliptic EquationsQingwei Li and Wenjie GaoAbstract. This paper is concerned with the existence of solutions to thefollowing singular elliptic boundary value problem involving p-Laplaceoperator?div(|?u| p?2 ?u) =hu γin Ω, u > 0 in Ω, u = 0 on ?Ω.Here, Ω ? R N (N ≥ 3) is a bounded doma...

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    Mediterr. J. Math. 13 (2016), 4917–4927DOI 10.1007/s00009-016-0782-91660-5446/16/064917-11published onlineAugust 16, 2016c ? Springer International Publishing 2016Existence of Weak Solutions to a Class ofSingular Elliptic EquationsQingwei Li and Wenjie GaoAbstract. This paper is concerned with the existence of solutions to thefollowing singular elliptic boundary value problem involving p-Laplaceoperator−div(|∇u| p−2 ∇u) =hu γin Ω, u > 0 in Ω, u = 0 on ∂Ω.Here, Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary,and h is a positive L 1 function on Ω. A “compatibility condition” onthe couple (h(x),γ) is given for the problem to have at least one solution.More precisely, it is shown that the problem admits at least one solutionif and only if there exists a u 0 ∈ W1,p0(Ω) such that?Ω hu1−γ0dx < ∞.This generalizes a previous result obtained by Sun and Zhang (Calc VarPartial Dif f er Equ 49:909–922, 2014) who considered the case p = 2.Mathematics Subject Classif i cation. 35K55, 35J60, 35J70.Keywords. p-Laplace operator, singularity, existence, compatibility con-dition.1. IntroductionLet Ω ⊂ R N , N ≥ 3 be a bounded domain with smooth boundary ∂Ω. Inthis paper, we are interested in the following singular elliptic problem:???−Δ p u =hu γin Ω,u > 0 in Ω,u = 0 on ∂Ω,(E h,γ )where Δ p u = div(|∇u| p−2 ∇u) is the standard p-Laplace operator, p > 1,γ >1 are real numbers, and h ∈ L 1 (Ω) is a positive function (i.e., h(x) > 0 a.e.in Ω ). We call u a solution of (E h,γ ) if u ∈ W1,p0(Ω) satisf i es (E h,γ ) in thefollowing sense:?Ω|∇u| p−2 ∇u · ∇?dx =?Ωh?u γdx, ∀? ∈ W1,p0(Ω). (∗)The project is supported by NSFC (11271154, 11401252), by Science and Technology De-velopment Project of Jilin Province (20150201058NY, 20160520103JH) and by the projectof The Education Department of Jilin Province (2015-463).

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