Quantitative profile decomposition and stability for a nonlocal Sobolev inequality

Abstract

In this paper, we focus on studying the quantitative stability of the nonlocal Sobolev inequality given bySHL(∫RN(|x|−μ⁎|u|2μ⁎)|u|2μ⁎dx)12μ⁎≤∫RN|∇u|2dx,∀u∈D1,2(RN), where ⁎ denotes the convolution of functions, 2μ⁎:=2N−μN−2 and SHL are positive constants that depends solely on N and μ. For N≥3 and 0<μ<N, it is well-known that, up to translation and scaling, the nonlocal Sobolev inequality possesses a unique extremal function W[ξ,λ] that is positive and radially symmetric.Our research consists of three main parts. Firstly, we prove a result that provides quantitative stability of the nonlocal Sobolev inequality with the level of gradients. Secondly, we establish the stability of profile decomposition in relation to the Euler-Lagrange equation of the aforementioned inequality for nonnegative functions. Lastly, we investigate the quantitative stability of the nonlocal Sobolev inequality in the following form:‖∇u−∑i=1κ∇W[ξi,λi]‖L2≤C‖Δu+(1|x|μ⁎|u|2μ⁎)|u|2μ⁎−2u‖(D1,2(RN))−1, where the parameter region satisfies κ≥2, 3≤N<6−μ, μ∈(0,N) with 0<μ≤4, or in the case of dimension N≥3 and κ=1, μ∈(0,N) with 0<μ≤4.