Existence of Weak Solutions to a Class of Singular Elliptic Equations
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- 作者：Qingwei Li ; Wenjie Gao
- 关键词：\({p}\) ; Laplace operator ; singularity ; existence ; compatibility condition
- 刊名：Mediterranean Journal of Mathematics
- 出版年：2016
- 出版时间：December 2016
- 年：2016
- 卷：13
- 期：6
- 页码：4917-4927
- 全文大小：516 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Mathematics - 出版者：Birkh盲user Basel
- ISSN：1660-5454
- 卷排序：13
文摘
This paper is concerned with the existence of solutions to the following singular elliptic boundary value problem involving \({p}\)-Laplace operator$$-{\rm div}(|\nabla u|^{p-2} \nabla u)=\frac{h}{u^\gamma} \text{in } \Omega, \quad u > 0\text{ in } \Omega,\quad u=0 \text{ on } \partial\Omega.$$Here, \({\Omega\subset \mathbb{R}^N(N\geq3)}\) is a bounded domain with smooth boundary, and \({h}\) is a positive \({L^1}\) function on \({\Omega}\). A “compatibility condition” on the couple \({(h(x),\gamma)}\) is given for the problem to have at least one solution. More precisely, it is shown that the problem admits at least one solution if and only if there exists a \({u_0\in W_0^{1,p}(\Omega)}\) such that \({\int_\Omega hu_0^{1-\gamma} \mathrm{d}x < \infty}\). This generalizes a previous result obtained by Sun and Zhang (Calc Var Partial Differ Equ 49:909–922, 2014) who considered the case \({p=2}\).Keywords\({p}\)-Laplace operatorsingularityexistencecompatibility conditionThe project is supported by NSFC (11271154, 11401252), by Science and Technology Development Project of Jilin Province (20150201058NY, 20160520103JH) and by the project of The Education Department of Jilin Province (2015-463).