Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space

Abstract

We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \begin{equation*} -{\rm div} A(x,\nabla u)= f\in L^1(Ω), \end{equation*} on a Lipschitz bounded domain in ${\mathbb{R}}^N$. The growth of the monotone vector field $A$ is controlled by a generalized nonhomogeneous and anisotropic $N$-function $M$. The approach does not require any particular type of growth condition of $M$ or its conjugate $M^{\ast }$ (neither ${\Delta }_2$, nor ${\nabla }_2$). The condition we impose is log-Holder continuity of $M$, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.