Existence of renormalized solutions for a nonlinear elliptic equation in Musielak framework and L1 data

Abstract

In this paper, we prove existence result of renormalized solutions in the setting of Musielak-Orlicz spaces $W^1_0L_\varphi(\Omega)$ for the following strongly nonlinear Dirichlet problem

$$A(u)+g(x,u,\nabla u)=f\quad \textrm{in }\Omega,$$

where $A$ is a Leray-Lions operator acting from its domain $D(A)\subset W^1_0L_\varphi(\Omega)$ into its dual, while $g(x, u,\nabla u)$ is a nonlinear term having a growth conditions with respect only to $\nabla u$, and does not satisfy any sign condition. The right-hand side $f$ belongs to $ L^1(\Omega)$. 
A modular-inequality of Poincar\'e type in this setting is also proved (Lemma 2.5).