Entropy solution for strongly nonlinear elliptic problems with lower order terms and $L^1$-data

Abstract

We give an existence result for strongly nonlinear elliptic equations of the type $$ -{\rm div}\Big(a(x,u,\nabla u)+\Phi(u)\Big)+g(x,u,\nabla u)+H(x,\nabla u) = f \text{ in } \Omega, $$where the right hand side $f$ belongs to $L^1(\Omega)$, $- {\rm div}(a(x,u,\nabla u))$is a Leray--Lions type operator with growth $|\nabla u|^{p-1}$ in $\nabla u$ and $\Phi\in C^0(\mathbb{R},\mathbb{R}^N)$. The critical growth condition on  $g$ is with respect to $\nabla u$ and no growth condition with respect to $u$, while the function $H(x,\nabla u)$ grows as $|\nabla u|^{p-1}$.