Electron. J. Diff. Equ., Vol. 2012 (2012), No. 111, pp. 1-16.

Multiplicity of positive solutions for quasilinear elliptic p-Laplacian systems

Asadollah Aghajani, Jamileh Shamshiri

Abstract:
We study the existence and multiplicity of solutions to the elliptic system
$$\displaylines{ -\hbox{div}(|\nabla u|^{p-2} \nabla u)+m_1(x)|u|^{p-2}u =\lambda g(x,u) \quad x\in \Omega,\cr -\hbox{div}(|\nabla v|^{p-2} \nabla v)+m_2(x)|v|^{p-2}v=\mu h(x,v) \quad x\in \Omega,\cr |\nabla u|^{p-2}\frac{\partial u}{\partial n}=f_u(x,u,v),\quad |\nabla v|^{p-2}\frac{\partial v}{\partial n}=f_v(x,u,v), }$$
where $\Omega\subset \mathbb{{R}}^N$ is a bounded and smooth domain. Using fibering maps and extracting Palais-Smale sequences in the Nehari manifold, we prove the existence of at least two distinct nontrivial nonnegative solutions.

Submitted June 19, 2012. Published July 2, 2012.
Math Subject Classifications: 35B38, 34B15, 35J92.
Key Words: Critical points; nonlinear boundary value problems; quasilinear p-Laplacian problem; fibering map; Nehari manifold.

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Asadollah Aghajani
School of Mathematics
Iran University of Science and Technology
P.O. Box 16846-13114, Narmak, Tehran, Iran
email: aghajani@iust.ac.ir
Jamileh Shamshiri
Department of Mathematics, Karaj Branch
Islamic Azad University, Karaj, Iran
email: jamileshamshiri@gmail.com

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