On critical p-Laplacian systems

الأصل · EN

We consider the critical p-Laplacian system equation92 cases-Δₚ u-λa/p|u|ᵃ⁻²u|v|ᵇ =μ₁|u|p-2u+αγ/p|u|α⁻²u|v|β, &x∈Ω, -Δₚ v-λb/p|u|ᵃ|v|ᵇ⁻²v =μ₂|v|p-2v+βγ/p|u|α|v|β⁻²v, &x∈Ω, u,vin D₀¹,ᵖ(Ω), cases equation where Δₚ:=div(|∇ u|ᵖ⁻²∇ u) is the p-Laplacian operator defined on D¹,ᵖ(Rⁿ):={u∈ Lp(Rⁿ):|∇ u|∈ Lᵖ(Rⁿ)}, endowed with norm uD¹,ᵖ:=(∫ᵣₙ|∇ u|ᵖdx)¹/ᵖ, N≥3, 1<p<N, λ, μ₁, μ₂≥ 0, γ≠0, a, b, α, β> 1 satisfy a + b = p, α+ β= p:=Np/N-p, the critical Sobolev exponent, Ω is Rⁿ or a bounded domain in Rⁿ, D₀¹,ᵖ(Ω) is the closure of C₀∞(Ω) in D¹,ᵖ(Rⁿ). Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution. We also consider the existence and multiplicity of nontrivial nonnegative solutions.