# Fully anisotropic elliptic problems with minimally integrable data

## TL;DR

This paper studies nonlinear anisotropic elliptic problems with minimal data assumptions, establishing existence, uniqueness, and regularity of solutions in complex Orlicz-Sobolev spaces without standard growth conditions.

## Contribution

It introduces a framework for solving anisotropic elliptic problems with general N-functions, extending existence and regularity results to minimally integrable data in non-reflexive Orlicz spaces.

## Key findings

- Existence and uniqueness of weak solutions under minimal integrability conditions.
- Generalized solutions exist even with measure data.
- Solutions exhibit maximal regularity in Marcinkiewicz spaces.

## Abstract

We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic $N$-function, which is not necessarily of power type and need not satisfy the $\Delta_2$ nor the $\nabla _2$-condition. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solutions - in the approximable sense - is established. Their maximal regularity in Marcinkiewicz-type spaces is exhibited as well. Uniqueness of approximable solutions is also proved in case of $L^1$-data.