A blow-up approach for singular elliptic problems with natural growth in the gradient

Abstract: We prove existence and nonexistence results concerning elliptic problems whose basic model is \begin{equation*} \begin{cases} \displaystyle-\Delta u+\mu(x)\frac{|\nabla u|^2}{(u+\delta)^\gamma}= \lambda u^p, &x\in \Omega, \ u> 0, &x\in \Omega, \ u=0, &x\in\partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N (N\geq 3)$ is a bounded smooth domain, $\lambda>0$, $p>1$, $\delta\geq 0$, $\gamma>0$ and $\mu\in L^\infty(\Omega)$. The main achievement resides in handling a possibly singular ($\delta=0$) first order term having a nonconstant coefficient $\mu$ in the presence of a superlinear zero order term. Our approach for the existence results is based on fixed point theory. With the aim of applying it, a previous analysis on a related non-homogeneous problem is carried out. The required a priori estimates are proven via a blow-up method.