Bulletin of the
Korean Mathematical Society BKMS

In this paper we study the existence of non-trivial weak solutions of the Neumann problem for quasilinear elliptic equations in the form $$ -\text{div}(h(x)|\nabla u|^{p-2}\nabla u)+b(x)|u|^{p-2}u= f(x,u) ,\quad p\geq 2 $$ in an unbounded domain $\Omega \subset {\mathbb R}^N, N\geq 3$, with sufficiently smooth bounded boundary $\partial\Omega$, where $h(x) \in L^1_{loc}(\overline \Omega)$, $\overline\Omega =\Omega \cup\partial\Omega$, $h(x)\geq 1 $ for all $x \in \Omega.$ The proof of main results rely essentially on the arguments of variational method.

Keywords: Neumann problem, $p$-Laplacian, Mountain pass theorem, the weakly continuously differentiable functional

MSC numbers: 35J20, 35J65