On critical p-Laplacian systems

2015/08/25 by Zhenyu Guo, Guo, Zhenyu, Kanishka Perera +3
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1508.06006

Abstract

We consider the critical p-Laplacian system \begincases-Δ_p u-(λa)/(p)|u|^a-2u|v|^b =μ₁|u|^{p^∗-2}u+(αγ)/(p^∗)|u|^α-2u|v|^β, amp;x∈Ω,
-Δ_p v-(λb)/(p)|u|^a|v|^b-2v =μ₂|v|^{p^∗-2}v+(βγ)/(p^∗)|u|^α|v|^β-2v, amp;x∈Ω,
u,v in D₀^1,p(Ω), \endcases where Δ_p:=div(|∇ u|^p-2∇ u) is the p-Laplacian operator defined on D^1,p(ℝ^N):=\u∈ L^{p^∗}(ℝ^N):|∇ u|∈ L^p(ℝ^N)\, endowed with norm ‖u‖_D^1,p:=(∫_ℝ^N|∇ u|^pdx)^(1)/(p), N≥3, 1 1 satisfy a + b = p, α+ β= p^∗:=(Np)/(N-p), the critical Sobolev exponent, Ω is ℝ^N or a bounded domain in ℝ^N, D₀^1,p(Ω) is the closure of C₀^∞(Ω) in D^1,p(ℝ^N). 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.

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