$T$-$p(x)$-solutions for nonlinear elliptic equations with an $L^{1}$-dual datum
Volume 39 / 2012
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 $\prod_{i=1}^{N}L^{p'(\cdot)} (\varOmega)$.
