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Existence of weak and renormalized solutions of degenerated elliptic equation

  • Benali AharrouchEmail author
  • Jaouad Bennouna
  • Bouchra El hamdaoui
Article
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Abstract

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 :
$$\begin{aligned} {\left\{ \begin{array}{ll} r(x)>\frac{N}{p(x)-1}, r(x)\ge p'(x)&{}\quad \forall x \in \Omega ,\\ q(x)>\max \left( \frac{N}{p(x)},1\right) , q(x)\ge p'(x)&{}\quad \forall x \in \Omega . \end{array}\right. } \end{aligned}$$

Keywords

Sobolev spaces with variable exponents Renormalized solutions Weak solution Quasilinear elliptic equations Boundary value problems 

Mathematics Subject Classification

35J70 35J67 46E35 

Notes

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Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El MahrazUniversity of FezAtlas FezMorocco

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