$T$-$p(x)$-solutions for nonlinear elliptic equations with an $L^{1}$-dual datum

Volume 39 / 2012

El Houssine Azroul, Abdelkrim Barbara, Meryem El Lekhlifi, Mohamed Rhoudaf Applicationes Mathematicae 39 (2012), 339-364 MSC: 35J60, 35J66, 46E35. DOI: 10.4064/am39-3-8

Abstract

We establish the existence of a $T$-$p(x)$-solution for the $p(x)$-elliptic problem $$ -{\rm div} (a(x,u,\nabla u))+g(x,u)=f-{\rm div} F\quad\mbox{ in } \varOmega, $$ where $\varOmega$ is a bounded open domain of $\mathbb{R}^{N}$, $N\geq 2$ and $a:\varOmega\times \mathbb{R}\times \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but with only a weak monotonicity condition. The right hand side $f$ lies in $L^1(\varOmega)$ and $F$ belongs to $\prod_{i=1}^{N}L^{p'(\cdot)} (\varOmega)$.

Authors

  • El Houssine AzroulLaboratory LAMA, Department of Mathematics
    Faculty of Sciences, Dhar-Mahraz
    B.P. 1796, Atlas Fez, Morocco
    e-mail
  • Abdelkrim BarbaraLaboratory LAMA, Department of Mathematics
    Faculty of Sciences, Dhar-Mahraz
    B.P. 1796, Atlas Fez, Morocco
    e-mail
  • Meryem El LekhlifiLaboratory LAMA, Department of Mathematics
    Faculty of Sciences, Dhar-Mahraz
    B.P: 1796
    Atlas Fez, Morocco
    e-mail
  • Mohamed RhoudafFaculty of Science and Technology
    Ziaten, km 10, old airport road
    B.P. 416 Tangier, Morocco
    e-mail

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