# An eigenvalue problem for the anisotropic $\Phi$-Laplacian

## TL;DR

This paper investigates an eigenvalue problem for a fully anisotropic elliptic operator within Orlicz-Sobolev spaces, extending the theory to non-radial, non-polynomial growth functions without the $ riangle_2$-condition.

## Contribution

It develops new theoretical aspects of anisotropic Orlicz-Sobolev spaces to analyze eigenvalue problems with general anisotropic N-functions.

## Key findings

- Established existence of eigenvalues for the anisotropic $\Phi$-Laplacian.
- Extended the functional framework to include non-radial and non-polynomial growth conditions.
- Developed new tools in anisotropic Orlicz-Sobolev space theory.

## Abstract

We study an eigenvalue problem involving a fully anisotropic elliptic differential operator in arbitrary Orlicz-Sobolev spaces. The relevant equations are associated with constrained minimization problems for integral functionals depending on the gradient of competing functions through general anisotropic $N$-functions. In particular, the latter need neither be radial, nor have a polynomial growth, and are not even assumed to satisfy the so called $\Delta_2$-condition. The resulting analysis requires the development of some new aspects of the theory of anisotropic Orlicz-Sobolev spaces.