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Article Dans Une Revue Mathematical Modelling and Analysis Année : 2024

Singular nonlinear problems with natural growth in the gradient

Boussad Hamour

Résumé

\begin{abstract} In this paper we consider the following problem:\\ $ \left\{\begin{array}{ll} -\textrm{div}\,(a(x,u,Du)=H(x,u,Du)+\dfrac{a_{0}(x)}{|u|^{\theta}}+\chi_{\{u\neq 0\}}\,f(x)& \mbox{in } \Omega, \\ u=0 \quad\textrm{on } \partial\Omega, & \end{array} \right. $\\ where $\Omega$ is an open bounded subset of $\mathbb{R}^{N}$, $1 0$, $0<\theta\leq 1$, $\chi_{\{u\neq 0\}}$ 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}
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Dates et versions

hal-04521520 , version 1 (26-03-2024)

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Boussad Hamour. Singular nonlinear problems with natural growth in the gradient. Mathematical Modelling and Analysis, 2024, 29 (2), pp.367-386. ⟨10.3846/mma.2024.17948⟩. ⟨hal-04521520⟩
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