A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities

This article deals with existence of solutions to the following fractional p p -Laplacian system of equations: ( − Δ p ) s u = ∣ u ∣ p s * − 2 u + γ α p s * ∣ u ∣ α − 2 u ∣ v ∣ β in Ω , ( − Δ p ) s v = ∣ v ∣ p s * − 2 v + γ β p s * ∣ v ∣ β − 2 v ∣ u ∣ α in Ω , \{{array}{l}{(-{Δ }ₚ)}ˢu={| u| }^{{p}ₛ*-2}u+{γ α }{{p}ₛ*}{| u| }α -2u{| v| }β{0.33em}{0.33em}{0.1em}in{0.1em}{0.33em}Ω ,{1.0em}\\ {(-{Δ }ₚ)}ˢv={| v| }^{{p}ₛ*-2}v+{γ β }{{p}ₛ*}{| v| }β -2v{| u| }α{0.33em}{0.33em}{0.1em}in{0.1em}{0.33em}Ω ,{1.0em}{array}. where s ∈ ( 0 , 1 ) s∈ (0,1) , p ∈ ( 1 , ∞ ) p∈ (1,∞ ) with N > s p N> sp , α , β > 1 α ,β > 1 such that α + β = p s * ≔ N p N − s p α +β ={p}ₛ*:= Np/N-sp and Ω = R N Ω ={{R}}N or smooth bounded domains in R N {{R}}N . When Ω = R N Ω ={{R}}N and γ = 1 γ =1 , we show that any ground state solution of the aforementioned system has the form ( λ U , τ λ V ) (λ U,τ λ V) for certain τ > 0 τ > 0 and U U and V V are two positive ground state solutions of ( − Δ p ) s u = ∣ u ∣ p s * − 2 u {(-{Δ }ₚ)}ˢu={| u| }^{{p}ₛ*-2}u in R N {{R}}N . For all γ > 0 γ > 0 , we establish existence of a positive radial solution to the aforementioned system in balls. When Ω = R N Ω ={{R}}N , we also establish existence of positive radial solutions to the aforementioned system in various ranges of γ γ .