Nonlinear eigenvalue problems for a biharmonic operator in Orlicz-Sobolev spaces

TL;DR

This paper introduces a new higher-order Laplacian operator within Orlicz-Sobolev spaces, explores its properties, and analyzes the eigenvalues and spectra of nonlinear eigenvalue problems related to this operator.

Contribution

It defines a generalized biharmonic g-Laplacian operator in Orlicz-Sobolev spaces, establishes its fundamental properties, and investigates the spectral behavior of associated nonlinear eigenvalue problems.

Findings

Eigenvalues depend on normalization due to operator non-homogeneity.

Spectra can concentrate at zero, infinity, or cover positive real numbers.

Basic functional properties facilitate existence results.

Abstract

In this paper, we introduce a new higher-order Laplacian operator in the framework of Orlicz-Sobolev spaces, the biharmonic g-Laplacian $$\Delta_g^2 u:=\Delta \left(\dfrac{g(|\Delta u|)}{|\Delta u|} \Delta u\right),$$ where $g=G'$, with $G$ an N-function. This operator is a generalization of the so called bi-harmonic Laplacian $\Delta^2$. Here, we also established basic functional properties of $\Delta_g^2$, which can be applied to existence results. Afterwards, we study the eigenvalues of $\Delta_g^2$, which depend on normalisation conditions, due to the lack of homogeneity of the operator. Finally, we study different nonlinear eigenvalue problems associated to $\Delta_g^2$ and we show regimes where the corresponding spectrum concentrate at $0$, $\infty$ or coincide with $(0, \infty)$.