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

Abstract

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 $\lambda.$

Keywords

• singular elliptic problems| positive solutions| bifurcation problems| sub and supersolutions| fixed points| multiplicity theorems