Multiplicity of positive solutions for quasilinear elliptic p-Laplacian systems

Abstract

We study the existence and multiplicity of solutions to the elliptic system $$displaylines{ -hbox{div}(|abla u|^{p-2} abla u)+m_1(x)|u|^{p-2}u =lambda g(x,u) quad xin Omega,cr -hbox{div}(|abla v|^{p-2} abla v)+m_2(x)|v|^{p-2}v=mu h(x,v) quad xin Omega,cr |abla u|^{p-2}frac{partial u}{partial n}=f_u(x,u,v),quad |abla v|^{p-2}frac{partial v}{partial n}=f_v(x,u,v), }$$ where $Omegasubset mathbb{{R}}^N$ 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.

Keywords

• Critical points
• nonlinear boundary value problems
• quasilinear p-Laplacian problem
• fibering map
• Nehari manifold