EXISTENCE OF INFINITELY MANY EIGENGRAPH SEQUENCES OF THE p(⋅)-BIHARMONIC EQUATION

Abstract

The aim of this paper is to study the nonlinear eigenvalue problem$$\begin{array}{r}(P)\{\begin{array}{l}\mathrm{\Delta }(|\mathrm{\Delta }u{|}^{p(x)-2}\mathrm{\Delta }u)-\lambda \zeta (x)|u{|}^{\alpha (x)-2}u=\mu \xi (x)|u{|}^{\beta (x)-2}u,x\in \mathrm{\Omega },u\in W^{2,p(\cdot )}(\mathrm{\Omega })\cap W_0^{1,p(\cdot )}(\mathrm{\Omega }),\end{array}\end{array}$$where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$, with smooth boundary $\partial \mathrm{\Omega }$, $N\ge 1$, $\lambda ,\mu$ are real parameters,$\zeta$ and $\xi$ are nonnegative functions, $p,\alpha$,and $\beta$ are continuousfunctions on $\bar{\mathrm{\Omega }}$ such that$1<\alpha (x)<\beta (x)<p(x)<\frac{N}{2}.$We show that the $p(\cdot )$-biharmonic operator possesses infinitelymany eigengraph sequences and also prove that the principal eigengraph exists.Our analysis mainly relies on variational method and we prove Ljusternik-Schnirelemanntheory on $C^1$-manifold.