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, $ $u \gt 0$ in $\Omega, $ where $\Omega$ is a bounded domain in $\mathbb{R}.{n}$ with $C.{2}$ boundary, $0\leq a\in L.{\infty}\left( \Omega\right), $ $0 \lt \alpha \lt 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 \gt 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 $\lambda \gt \Lambda.$ 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 $\lambda.$