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

In this paper, we study a higher‐order Laplacian operator in the framework of Orlicz–Sobolev spaces, the biharmonic g‐Laplacian Δg2u:=Δg(|Δu|)|Δu|Δu,$$\begin{equation*} \Delta _g^2 u:=\Delta {\left(\dfrac{g(|\Delta u|)}{|\Delta u|} \Delta u\right)}, \end{equation*}$$where g=G′$g=G^{\prime }$, with G$G$ being an N‐function. This operator is a generalization of the so‐called bi‐harmonic Laplacian Δ2$\Delta ^2$. Here, we also establish basic functional properties of Δg2$\Delta _g^2$, which can be applied to existence results. Afterwards, we study the eigenvalues of Δg2$\Delta _g^2$, which depend on normalization conditions, due to the lack of homogeneity of the operator. Finally, we study different nonlinear eigenvalue problems associated to Δg2$\Delta _g^2$ and we show regimes where the corresponding spectrum concentrates at 0, ∞$\infty$ or coincide with (0,∞)$(0, \infty)$.