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.