Multiple finite-energy positive weak solutions to singular elliptic problems with a parameter

AIMS Mathematics

关键词: : ;    singular elliptic problems;    positive solutions;    bifurcation problems;    sub and supersolutions;    fixed points;    multiplicity theorems;   

DOI  :  10.3934/Math.2018.1.233

学科分类：数学（综合）

来源: AIMS Press

【 摘 要 】

Consider the problem $-\Delta u=a\left(x\right)u^{-\alpha}+f\left(\lambda,x,u\right)$ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u>0$ in$\Omega,$ \ where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $C^{2}$ boundary, $0\leqa\in L^{\infty}\left(\Omega\right),$ $0<\alpha<3 and="" f="" left="" lambda="" x="" right="" is="" nonnegative="" and="" superlinear="" with="" subcritical="" growth="" at="" infty="" we="" prove="" that="" if="" f="" satisfies="" some="" additional="" conditions="" then="" for="" some="" lambda="">0 ,$ there are at least two weak solutions in$H_{0}^{1}\left(\Omega\right)\cap C\left(\overline{\Omega}\right)$ if$\lambda\in\left(0,\Lambda\right)$, and there is no weak solution in$H_{0}^{1}\left(\Omega\right)\cap L^{\infty}\left(\Omega\right)$ if$\lambda>\Lambda.$ We also prove that, for each $\lambda\in\left[0,\Lambda\right] $, there exists a unique minimal weak solution $u_{\lambda}$ in $H_{0}^{1}\left(\Omega\right)\cap L^{\infty}\left(\Omega\right) $, which is strictly increasing in $\lambda.$.

【 授权许可】

CC BY