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-Δu+μ(x)\frac{|\nabla u|^2}{(u+δ)^γ}= λu^p, &x\in Ω, \\ u> 0, &x\in Ω, \\ u=0, &x\in\partialΩ, \end{cases} \end{equation*} where $\Omega \subset {\mathbb{R}}^N(N\ge 3)$ is a bounded smooth domain, $\lambda >0$, $p>1$, $\delta \ge 0$, $\gamma >0$ 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.