On Dirichlet problems with singular nonlinearity of indefinite sign

Abstract

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq1$, let $K$, $M$ be two nonnegative functions and let $\alpha,\gamma>0$. We study existence and nonexistence of positive solutions for singular problems of the form $-\Delta u=K\left( x\right) u^{-\alpha}-\lambda M\left( x\right) u^{-\gamma}$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\lambda>0$ is a real parameter. We mention that as a particular case our results apply to problems of the form $-\Delta u=m\left( x\right) u^{-\gamma}$ in $\Omega$, $u=0$ on $\partial\Omega$, where $m$ is allowed to change sign in $\Omega$.