The Nehari manifold for systems of nonlinear elliptic equations

Abstract

This paper deals with existence and multiplicity results of nonlocal positive solutions to the following system −∆pu = λ|u|p1−2u+ (α+ 1)u|u|α−1|v|β+1, −∆qv = µ|v|q−2v + (β + 1)|u|α+1|v|β−1v, together with Dirichlet or mixed boundary conditions, under some hypotheses on the parameters p, p1, α, β and q. More precisely, the system considered corresponds to a perturbed eigenvalue equation combined with a second equation having concave and convex nonlinearities. The study is based on the extraction of Palais-Smale sequences in the Nehari manifold. The behaviour of the energy corresponding to these positive solutions, with respect to the real parameters λ and µ, is established