Existence of solutions for a fourth order eigenvalue problem with variable exponent under Neumann boundary conditions

In this work we will study the eigenvalues for a fourth order elliptic equation with $p(x)$-growth conditions $\Delta^2_{p(x)} u=\lambda |u|^{p(x)-2} u$, under Neumann boundary conditions, where $p(x)$ is a continuous function defined on the bounded domain with $p(x)>1$. Through the Ljusternik-Schnireleman theory on $C^1$-manifold, we prove the existence of infinitely many eigenvalue sequences and $\sup \Lambda =+\infty$, where $\Lambda$ is the set of all eigenvalues.