Multiple positive solutions for a critical fractional <i>p</i> -Laplacian system with concave nonlinearities

Abstract

The aim of this paper is to study the following nonlinear fractional p-Laplacian system with critical exponents: {(−Δ)psu+|u|p−2u=λg(x)|u|q−2u+αα+βf(x)|u|α−2u|v|β in Ω,(−Δ)psv+|v|p−2v=μh(x)|v|q−2v+βα+βf(x)|u|α|v|β−2v in Ω,u=v=0 in RN∖Ω, where Ω is a smooth bounded set in RN, 0<s<1, λ,μ>0 are two parameters, 1<q<p<ps∗, N>ps, α,β>1 satisfy α+β=ps∗ with ps∗=npn−ps is the fractional Sobolev critical exponent and (−Δ)ps is the fractional p-Laplacian operator. Using the Nehari manifold and Ljusternik–Schnirelmann category, we study the topology of the global maximum set Θ of f(x), and show that the system has at least at least catΘδ(Θ)+1 distinct positive solutions.