On existence, non-existence and blow-up results for a singular semilinear Laplacian problem

Abstract

We study the optimal value of p for solvability of the problem

(1)

Here $\lambda ,\alpha >0$, $p>1$, f is a non-negative measurable function and , $N\geqslant 3$, is an open bounded domain with smooth boundary such that $0 \in \mathrm{\Omega }$. We find the critical threshold exponent $p_{+}(\lambda ,\alpha )$ for solvability of (1) and show that if , $1<p<p_{+}(\lambda ,\alpha )$ and for some sufficiently small $c_0>0$, then there exists a solution as a limit of solutions to approximating problems. Moreover, for $p \geqslant p_{+}(\lambda ,\alpha )$ we show that a complete blow-up phenomenon occurs.