Renormalized solution for nonlinear elliptic problems with lower order terms and L1 data in Musielak-Orlicz spaces

Abstract

We prove the existence of a renormalized solution for the nonlinear elliptic problem $$-\textrm{div}\ a(x,u,\nabla u)-\textrm{div}\ \phi(u)+g(x,u,\nabla u)=f \textrm{ in } \Omega,$$ in the setting of Musielak-Orlicz spaces. $\phi\in\mathcal{C}^{0}(\mathbb{R},\mathbb{R}^{N}),$ the nonlinearity $g$ has a natural growth with respect to its third argument and satisfies the sign condition while the datum $f$ belongs to $L^1(\Omega).$ No $\Delta_2$-condition is assumed on the Musielak function.