Existence of entropy solutions for nonlinear elliptic equations in Musielak framework with L1 data

Abstract

We prove existence of solutions for strongly nonlinear elliptic equations of the form $$ \left\{\begin{array}{c} A(u)+g(x,u,\nabla u)=f+\mbox {div}(\phi(u))\quad \textrm{in }\Omega, \\ u\equiv0\quad \partial \Omega. \end{array} \right.$$ Where $A(u)=-\mbox {div}(a(x,u,\nabla u))$ be a Leray-Lions operator defined in $D(A)\subset W^{1}_{0}L_\varphi(\Omega) \rightarrow W^{-1}_{0}L_\psi(\Omega)$, the right hand side belongs in $ L^{1}(\Omega)$, and $\phi\in C^{0}(\mathbb{R},\mathbb{R}^N)$, without assuming the $\Delta_{2}$-condition on the Musielak function.