The minimizing problem involving p--Laplacian and Hardy--Littlewood--Sobolev upper critical exponent

2018/05/28 by Su, Yu, Chen, Haibo
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.10986

Abstract

In this paper, we study the minimizing problem: S_p,1,α,μ:= inf_u∈ W^1,p(ℝ^N)∖\0\ \frac ∫_ℝ^N|∇ u|^pdx - μ∫_ℝ^N \frac|u|^p|x|^p dx ( ∫_ℝ^N ∫_ℝ^N \frac|u(x)|^p^*_α|u(y)|^p^*_α|x-y|^α dx dy )^\fracp2⋅ p^*_α, where N\geqslant3, p∈(1,N), μ∈ [ 0, ( (N-p)/(p) )^p ), α∈(0,N) and p^*_α= (p)/(2)((2N-α)/(N-p)) is the Hardy--Littlewood--Sobolev upper critical exponent. Firstly, by using refinement of Hardy-Littlewood-Sobolev inequality, we prove that S_p,1,α,μ is achieved in ℝ^N by a radially symmetric, nonincreasing and nonnegative function. Secondly, we give a estimation of extremal function.

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