Existence of weak and renormalized solutions of degenerated elliptic equation

Autor: Jaouad Bennouna, Benali Aharrouch, Bouchra El hamdaoui
Rok vydání: 2019
Předmět:
Zdroj: Afrika Matematika. 30:755-776
ISSN: 2190-7668
1012-9405
DOI: 10.1007/s13370-019-00682-3
Popis: We consider the degenerate nonlinear elliptic equation (E) : $${\mathcal {A}}(u)= g-{\text {div}}(f)$$ , where $${\mathcal {A}}(u)=-{\text {div}}(a(x,u,\nabla u))$$ is a Leray-Lions operator defined on $$W_0^{1,p(\cdot )}(\Omega )$$ allowed to be non linear degenerated. The main gaol of this paper is to prove in first, an $$L^{\infty }(\Omega )$$ estimate for the bounded solution of (E), and then the existence of a weak and a renormalized solution of (E), with $$f\in (L^{r(\cdot )}(\Omega ))^N, g\in L^{q(\cdot )}(\Omega )$$ , where $$r(\cdot )$$ and $$q(\cdot )$$ satisfies the following conditions : $$\begin{aligned} {\left\{ \begin{array}{ll} r(x)>\frac{N}{p(x)-1}, r(x)\ge p'(x)&{}\quad \forall x \in \Omega ,\\ q(x)>\max \left( \frac{N}{p(x)},1\right) , q(x)\ge p'(x)&{}\quad \forall x \in \Omega . \end{array}\right. } \end{aligned}$$
Databáze: OpenAIRE
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