### Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities

#### Abstract

We study the existence of multiple positive solutions of $-\Delta u = \lambda u^{-q} +u^p$ in $\Omega$ with homogeneous Dirichlet boundary condition, where $\Omega$ is a bounded domain in $\mathbb R^N$, $\lambda >0$, and $0 < q \leq 1 < p \leq (N+2)/(N-2)$. We show by a variational method that if $\lambda$ is less than some positive constant then the problem has at least two positive, weak solutions including the cases of $q=1$ or $p=(N+2)/(N-2)$. We also study the regularity of positive weak solutions.

#### Article information

Source
Adv. Differential Equations, Volume 9, Number 1-2 (2004), 197-220.

Dates
First available in Project Euclid: 18 December 2012