Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 111, pp. 1-16.
Title: Multiplicity of positive solutions for quasilinear elliptic
p-Laplacian systems
Authors: Asadollah Aghajani (Iran Univ. of Science and Tech., Tehran, Iran)
Jamileh Shamshiri (Islamic Azad Univ., Karaj, Iran)
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 02, 2012.
Math Subject Classifications: 35B38, 34B15, 35J92.
Key Words: Critical points; nonlinear boundary value problems;
quasilinear p-Laplacian problem; fibering map; Nehari manifold.