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

Abstract

In this paper, we study a higher-order Laplacian operator in the framework of Orlicz–Sobolev spaces, the biharmonic g-Laplacian Δ2 𝑔𝑢 ∶= Δ(𝑔(|Δ𝑢|) |Δ𝑢| Δ𝑢) , where 𝑔=𝐺′ , with 𝐺 being an N-function. This operator is a generalization of the so-called bi-harmonic Laplacian Δ2. Here, we also establish basic functional properties of Δ2 𝑔, which can be applied to existence results. Afterwards, we study the eigenvalues of Δ2 𝑔, which depend on normalization conditions, due to the lack of homogeneity of the operator. Finally, we study different nonlinear eigenvalue problems associated to Δ2 𝑔 and we show regimes where the corresponding spectrum concentrates at 0, ∞ or coincide with (0, ∞).