On Dirichlet problems with singular nonlinearity of indefinite sign

Abstract

Let $\Omega$ be a smooth bounded domain in ${\mathbb{R}}^N$, $N\ge 1$, 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 $\Omega$.

To appear in Journal of Mathematical Analysis and Applications