Abstract

We will consider the following problem $$-\Delta u-\lambda \frac{u}{|x|^2}=|\nabla u|^p+c\,f,\quad u>0 \hbox{ in } \Omega,\quad $$ where $ \Omega \subset \mathbb{R}^N$ is a domain such that $0\in \Omega $, $N\ge 3$, $c>0$ and $\lambda >0$. The main objective of this note is to study the precise threshold $p_+=p_+$ for which there is no very weak supersolution if $ p\ge p_+$. The optimality of $p_+$ is also proved by showing the solvability of the Dirichlet problem when $1\le p 0$ small enough and $f\ge 0$ under some hypotheses that we will prescribe