On a Dirichlet problem in bounded domains with singular nonlinearity
Giuseppe Maria Coclite, Mario Michele Coclite · Discrete and Continuous Dynamical Systems · 2013
In this paper we prove the existence and regularity of positive solutions of the homogeneous Dirichlet problem\begin{equation*}-Δ u=g(x,u) in \Omega, u=0 on ∂ \Omega,\end{equation*}where $g(x,u)$ can be singular as $u\rightarrow0^+$ and $0\le g(x,u)\le\frac{\varphi_0(x)}{u^p}$ or $0\le$ $ g(x,u)$ $\le$ $\varphi_0(x)(1+\frac{1}{u^p})$, with $\varphi_0 \in L^m(\Omega), 1 ≤ m.$There are no assumptions on the monotonicity of $g(x,\cdot)$ and the existence of super- or sub-solutions.