Singular nonlinear problems with natural growth in the gradient
Résumé
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:\\ \indent $ -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 $\|a_{0}\|_{N/p}$ and $\|f\|_{N/p}$ sufficiently small, we prove the existence of at least one solution $u$ of this problem which is moreover such that the function $\exp(\delta|u|)-1 $ belongs to $W_{0}^{1,p}(\Omega)$ for some $\delta\geq \gamma$. This solution satisfies some a priori estimates in $W_0^{1,p}(\Omega)$. \end{abstract}