### On the summability of weak solutions for a singular Dirichlet problem in bounded domains

#### Abstract

In this paper, we prove the existence of positive weak solutions for the homogeneous Dirichlet problem associated to the equation \begin{equation*} -\Delta u=f(x,u)+\lambda h(x,u),\quad \text{in } \Omega, \end{equation*} where $\lambda\ge 0,$ $f(x,u)$ can be singular as $u \rightarrow0^+$ and $h(x,u)$ can diverge as $u \rightarrow\infty.$ We assume that $0\le f(x,u)\le\frac{\psi_0(x)}{u^p}$ with $\psi_0\in L^m(\Omega),\,\, m\ge 1,$ and $0\le h(x,u)\le\psi_\infty(x)u^q$ with $\psi_\infty\in L^M(\Omega),\,\, M\gt\frac{N}{2}.$ We do not have any monotonicity assumption on $f(x,\cdot)$, and $h(x,\cdot)$. Moreover, we do not assume the existence of any super or sub solution.

#### Article information

Source
Adv. Differential Equations, Volume 19, Number 5/6 (2014), 585-612.

Dates
First available in Project Euclid: 3 April 2014