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

Abstract

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

Theorems & Definitions (40)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Theorem 2.4
• Remark 2.5
• Proposition 2.6
• Lemma 2.7
• Definition 3.1
• Proposition 4.1
• proof