On Elliptic Systems involving critical Hardy-Sobolev exponents

2015/04/04 by Zhong, Xuexiu, Zou, Wenming
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1504.01005

Abstract

Let Ω⊂ \R^N (N≥ 3) be an open domain which is not necessarily bounded. By using variational methods, we consider the following elliptic systems involving multiple Hardy-Sobolev critical exponents: \begincases -Δu-λ\frac|u|^{2^*(s₁)-2}u|x|^s₁=κα\frac1|x|^s₂|u|^α-2u|v|^β amp;\hboxin Ω,
-Δv-μ\frac|v|^{2^*(s₁)-2}v|x|^s₁=κβ\frac1|x|^s₂|u|^α|v|^β-2v amp;\hboxin Ω,
(u,v)∈ \mathscrD:=D₀^1,2(Ω)× D₀^1,2(Ω), \endcases where s₁,s₂∈ (0,2), α>1,β>1, λ>0,μ>0,κ≠ 0, α+β≤ 2^*(s₂). Here, 2^*(s):=(2(N-s))/(N-2) is the critical Hardy-Sobolev exponent. We mainly study the critical case (i.e., α+β=2^*(s₂)) when Ω is a cone (in particular, Ω=\R₊^N or Ω=\R^N). We will establish a sequence of fundamental results including regularity, symmetry, existence and multiplicity, uniqueness and nonexistence, \it etc. In particular, the sharp constant and extremal functions to the following kind of double-variable inequalities S_α,β,λ,μ(Ω) (∫_Ω(λ\frac|u|^{2^*(s)}|x|^s+μ\frac|v|^{2^*(s)}|x|^s+2^*(s)κ(|u|^α|v|^β)/(|x|^s))dx)^{(2)/(2^*(s))} ≤ ∫_Ω(|∇ u|²+|∇ v|²)dx for (u,v)∈ \mathscrD will be explored. Further results about the sharp constant S_α,β,λ,μ(Ω) with its extremal functions when Ω is a general open domain will be involved.

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