A SINGULAR NONLINEAR PROBLEMS WITH NATURAL GROWTH IN THE GRADIENT

Abstract

In this paper, we consider the equation $-\textrm{div}\,(a(x,u,Du){=}H(x,u,Du)\\{+}\frac{a_{0}(x)}{\vert u \vert^{\theta}}+\chi_{\{u\neq 0\}}\,f(x)$ {in} $\Omega$, with boundary conditions $u=0$ {on} $\partial\Omega$, where $\Omega$ is an open bounded subset of $\mathbb{R}^{N}$, $1<p< N$, $-\mbox{div}(a(x,u,Du))$ is a Leray-Lions operator defined on $W_{0}^{1,p}(\Omega)$, $a_{0}\in L^{N/p}(\Omega )$, $a_{0}> 0$, $0<\theta\leq 1$, $\chi_{\{u\neq 0\}}$ is a characteristic function, $f\in L^{N/p}(\Omega)$ and $H(x,s,\xi)$ is a Carath\'eodory function such that $-c_{0}\, a(x,s,\xi)\xi\,\leq H(x,s,\xi)\,\mbox {sign}(s)\leq \gamma\,a(x,s,\xi)\xi \quad \mbox {a.e. } x\in \Omega , \forall s\in\mathbb{R}\,\, , \forall\xi \in \mathbb{R}^{N}. $ For $\Vert a_{0}\Vert_{N/p}$ and $\Vert f\Vert_{N/p}$ sufficiently small, we prove the existence of at least one solution $u$ of this problem which is moreover such that the function $\exp(\delta \vert u \vert)-1 $ belongs to $W_{0}^{1,p}(\Omega)$ for some $\delta\geq \gamma$. This solution satisfies some a priori estimates in $W_0^{1,p}(\Omega)$.