The Nehari manifold for systems of nonlinear elliptic equations

This paper deals with the existence and multiplicity results of nonlocal positive solutions to the following system: -Δpu=λ|u|p1-2u+(α+1)u|u|α-1|v|β+1,-Δqv=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 λλ and μμ, is established.