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

El Houssine Azroul ¹   ; Abdelkrim Barbara ¹   ; Meryem El Lekhlifi ²   ; Mohamed Rhoudaf ³

¹ Laboratory LAMA, Department of Mathematics Faculty of Sciences, Dhar-Mahraz B.P. 1796, Atlas Fez, Morocco
² Laboratory LAMA, Department of Mathematics Faculty of Sciences, Dhar-Mahraz B.P: 1796 Atlas Fez, Morocco
³ Faculty of Science and Technology Ziaten, km 10, old airport road B.P. 416 Tangier, Morocco

Applicationes Mathematicae, Tome 39 (2012) no. 3, pp. 339-364

Résumé

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 $\prod_{i=1}^{N}L^{p'(\cdot)} (\varOmega)$.

Détail

DOI : 10.4064/am39-3-8

Keywords: establish existence t p solution elliptic problem div nabla f div quad mbox varomega where varomega bounded domain mathbb geq varomega times mathbb times mathbb rightarrow mathbb carath odory function satisfying natural growth condition coercivity condition only weak monotonicity condition right side lies varomega belongs prod cdot varomega