\(T-p(x)\)-solutions for nonlinear elliptic equations with an \(L^{1}\)-dual datum (Q2913203)

The authors consider the existence of a \(T-p(x)\)-solution for the \(p(x)\)-elliptic problem NEWLINE\[NEWLINE -\text{div} (a(x,u,\nabla u))+g(x,u)=f-\text{div} F\quad\text{ in } \varOmega, NEWLINE\]NEWLINE 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 \(\prod_{i=1}^{N}L^{p'(\cdot)} (\varOmega)\).