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

We prove existence and nonexistence results concerning elliptic problems whose basic model is \(\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\mu (x)\frac{|\nabla u|^2}{(u+\delta )^\gamma }= \lambda u^p, & x\in \Omega ,\\ u> 0, & x\in \Omega ,\\ u=0, & x\in \partial \Omega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle -\mathrm{\Delta }u+\mu (x)\frac{{|\mathrm{\nabla }u|}^2}{{(u+\delta )}^{\gamma }}=\lambda u^p,}\hfill & x\in \mathrm{\Omega },\hfill \\ u>0,\hfill & x\in \mathrm{\Omega },\hfill \\ u=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \subset \mathbb {R}^N (N\ge 3)\) $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 3)$ is a bounded smooth domain, \(\lambda >0\) $\lambda >0$ , \(p>1\) $p>1$ , \(\delta \ge 0\) $\delta \ge 0$ , \(\gamma >0\) $\gamma >0$ and \(\mu \in L^\infty (\Omega )\) $\mu \in L^{\infty }(\mathrm{\Omega })$ . The main achievement resides in handling a possibly singular ( \(\delta =0\) $\delta =0$ ) first order term having a nonconstant coefficient \(\mu \) $\mu$ in the presence of a superlinear ( \(p>1\) $p>1$ ) zero order term. Our approach for the existence results is based on a fixed point argument. As a first step of this argument, an analysis on a related nonhomogeneous singular problem is carried out. The required a priori estimates are proved via a blow-up method.