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 On existence of weak solutions of Neumann problem for quasilinear elliptic equations involving $p$-Laplacian in an unbounded domain Bull. Korean Math. Soc. 2011 Vol. 48, No. 6, 1169-1182 https://doi.org/10.4134/BKMS.2011.48.6.1169Published online November 1, 2011 Trinh Thi Minh Hang and Hoang Quoc Toan Hanoi University of Civil Engineering, Hanoi University of Science Abstract : 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 Downloads: Full-text PDF