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.2011.48.6.1169
Published 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
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