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

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 $\,\displaystyle 0\le f(x,u)\le\frac{\psi_0(x)}{u^p}\,$ with $\,\psi_0\in L^m(\Omega),\,\, 1\, \le m,\,$ and $\,0\le h(x,u)\le\psi_\infty(x)u^q\,$ with $\,\psi_\infty\in L^M(\Omega),\,\, 1\,\le M.\,$ 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.