Fractional p-Laplace systems with critical Hardy nonlinearities: Existence and multiplicity

Abstract

Let Ω⊂Rd be a bounded open set containing zero, s ∈ (0, 1) and p ∈ (1, ∞). In this paper, we first deal with the existence, non-existence and some properties of ground-state solutions for the following class of fractional p -Laplace systems {(−Δp)su=αq|u|α−2u|v|β|x|minΩ,(−Δp)sv=βq|v|β−2v|u|α|x|minΩ,u=v=0inRd∖Ω, where d > sp , α+β=q where p≤q≤ps*(m) where ps*(m)=p(d−m)d−sp with 0 ≤ m ≤ sp . Additionally, we establish a concentration-compactness principle related to this homogeneous system of equations. Next, the main objective of this paper is to study the following non-homogenous system of equations {(−Δp)su=η|u|r−2u+γαps*(m)|u|α−2u|v|β|x|minΩ,(−Δp)sv=η|v|r−2v+γβps*(m)|v|β−2v|u|α|x|minΩ,u=v=0inRd∖Ω, where η, γ > 0 are parameters and p≤r<ps*(0). Depending on the values of η, γ , we obtain the existence of a non semi-trivial solution with the least energy. Further, for m=0, we establish that the above problem admits at least catΩ(Ω) nontrivial solutions.