2025/03/09 by Lu, Qikai, Yang, Minbo, Zhao, Shunneng
#35J08 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35A23 #Secondly 35B35
paper · doi:10.48550/arxiv.2503.06636
In this paper, we study the following fractional nonlocal Sobolev-type inequality CHLS(∫ℝn(|x|-μ ∗ |u|ps)|u|ps dx)(1)/(ps)≤‖u‖_Hs(ℝn)2 for all~~u∈ Hs(ℝn), induced by the classical fractional Sobolev inequality and Hardy-Littlewood-Sobolev inequality for s∈(0,(n)/(2)), μ∈(0,n) and where ps=(2n-μ)/(n-2s)≥2 is energy-critical exponent. The CHLS>0 is a constant depending on the dimension n, parameters s and μ, which can be achieved by W(x), and up to translation and scaling, W(x) is the unique positive and radially symmetric extremal function of the nonlocal Sobolev-type inequality. It is well-known that, up to a suitable scaling, (-Δ)su=(|x|-μ∗ |u|ps)|u|ps-2u for all~~u∈Hs(ℝn), is the Euler-Lagrange equation corresponding to the associated minimization problem. In this paper, we first prove the non-degeneracy of positive solutions to the critical Hartree equation for all s∈(0,(n)/(2)), μ∈(0,n) with 0<μ≤4s. Furthermore, we show the existence of a gradient type remainder term and, as a corollary, derive the existence of a remainder term in the weak L(n)/(n-2s)-norm for functions supported in domains of finite measure, under the condition s∈(0,(n)/(2)). Finally, we establish a Struwe-type profile decomposition and quantitative stability estimates for critical points of the above inequality in the parameter region s∈(0,(n)/(2)) with the number of bubbles κ≥1, and for μ∈(0,n) with 0<μ≤4s. In particular, we provide an example to illustrate the sharpness of our result for n=6s and μ=4s.