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

## TL;DR

This paper establishes the existence of renormalized solutions for a broad class of nonlinear elliptic equations in Musielak-Orlicz spaces without growth restrictions, using advanced approximation and measure techniques.

## Contribution

It proves the existence of solutions in Musielak-Orlicz spaces under minimal growth assumptions, extending previous results to more general anisotropic and nonhomogeneous settings.

## Key findings

- Existence of renormalized solutions without growth restrictions.
- Applicability to general nonlinear elliptic equations in Musielak-Orlicz spaces.
- Use of truncation, Young measures, and monotonicity methods.

## 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(\Omega), \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 $M^*$ (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.