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

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