Journal of the
Korean Mathematical Society JKMS

This paper studies the eigenvalues of the $G(\cdot)$-Laplacian \linebreak Dirichlet problem $$\left \{ \begin{aligned} \displaystyle -\text{div}\left(\frac{g(x,|\nabla u|)}{|\nabla u|}\nabla u\right) & = \displaystyle \lambda \left(\frac{g(x,|u|)}{|u|}u\right) & &\text{in} \; \Omega,\\ u & = 0 & &\text{on} \; \partial\Omega, \end{aligned} \right.$$ where $\Omega$ is a bounded domain in $\mathbb R^N$ and $g$ is the density of a generalized $\Phi$-function $G(\cdot)$. Using the Lusternik-Schnirelmann principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues.

Keywords: Dirichlet eigenvalue problems, Lusternik-Schnirelmann principle, $G(\cdot)$-Laplacian, Musielak-Orlicz growth, generalized $\Phi$-function, generalized Orlicz-Sobolev space

MSC numbers: Primary 35P30, 47J10, 49R05