Necessary and sufficient conditions of existence for a system involving the p-Laplacian (1<p<N)

AbstractIn this work, we study the existence of positive solutions for the system(P)−Δpu=μf(x,u,v)inRN,−Δpv=μg(x,u,v)inRN,lim|x|→∞u(x)=0=lim|x|→∞v(x),where Δpu=div(|∇u|p−2∇u) is the p-Laplacian (1<p<N). We prove that for 0<μ⩽μ∗ (P) admits a positive solution and that (P) has no positive solution for μ>μ∗. In some case we can have μ∗=μ∗ because these parameters depend on p,N and the growth of f and g. The existence result is related to a sub,super-solution methods and the nonexistence result is a consequence of Picone's identity