EXISTENCE OF INFINITELY MANY EIGENGRAPH SEQUENCES OF THE $p(\cdot)$-BIHARMONIC EQUATION

Abstract

The aim of this paper is to study the nonlinear eigenvalue problem\begin{align*}(P)\quad\begin{cases}\Delta (|\Delta u|^{p(x)-2}\Delta u)-\lambda \zeta(x)|u|^{\alpha(x)-2} u=\mu \xi(x) |u|^{\beta(x)-2}u, \quad x\in\Omega, u\in W^{2,p(\cdot)}(\Omega)\cap W_0^{1,p(\cdot)}(\Omega),\end{cases}\end{align*}where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, with smooth boundary $\partial\Omega$, $N\geq 1$, $\lambda, \mu$ are real parameters,$\zeta$ and $\xi$ are nonnegative functions, $p, \alpha$,and $\beta$ are continuousfunctions on $ \overline{\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.