An eigenvalue problem for a generalized polyharmonic operator in Orlicz-Sobolev spaces without the $\Delta_2$-condition

TL;DR

This paper investigates a generalized polyharmonic eigenvalue problem in Orlicz-Sobolev spaces without the $ ext{Δ}_2$-condition, proving the existence of infinitely many eigenvalues and eigenfunctions, and establishing initial regularity results.

Contribution

It extends eigenvalue theory to a broad class of operators in Orlicz-Sobolev spaces without the $ ext{Δ}_2$-condition, introducing new existence and regularity results.

Findings

Infinitely many eigenvalues and eigenfunctions exist.

Eigenvalues tend to infinity.

First regularity result for eigenfunctions established.

Abstract

In this paper, we consider a generalized polyharmonic eigenvalue problem of the form $A(u)= \lambda h(u)$ in a bounded smooth domain with Dirichlet boundary conditions in the setting of higher-order Orlicz-Sobolev spaces. Here, $A$ is a very general operator depending on $u$ and arbitrary higher-order derivatives of $u$, whose growth is governed by an Orlicz function, and $h$ is a lower order term. Combining the theories of pseudomonotone operators with complementary systems, we prove that this eigenvalue problem has an infinite number of eigenfunctions and that the corresponding sequence of eigenvalues tends to infinity. We point out that the $\Delta_2$-condition is not assumed for the involved Orlicz functions. Finally, we prove a first regularity result for eigenfunctions by following a De Giorgi's iteration scheme.