| Full text | |
| Author(s): |
Piccione, Paolo
;
Yang, Minbo
;
Zhao, Shunneng
Total Authors: 3
|
| Document type: | Journal article |
| Source: | Journal of Differential Equations; v. 417, p. 41-pg., 2025-02-05. |
| Abstract | |
In this paper, we focus on studying the quantitative stability of the nonlocal Sobolev inequality given by S-HL(integral(N)(R) (|x|-(mu) * |mu|(2*)(mu) |mu|(2*)(mu) dx)1/(2*)(mu) <= integral(N)(R) | del u|(2) dx, for all u is an element of D-1,D-2(R-N) , where * denotes the convolution of functions, 2 * (mu) := 2 N-mu/ N- 2 and S HL are positive constants that depends solely on N and mu . For N> 3 and 0 < mu < N , it is well-known that, up to translation and scaling, the nonlocal Sobolev inequality possesses a unique extremal function W[ xi,lambda] 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: parallel to del u-(kappa)Sigma(i=1)del W[xi i,lambda i]parallel to parallel to parallel to(2)(L)<= C parallel to parallel to parallel to Delta u+(1|x|(mu)*|u|(2*)(mu))|u|(2*;)(mu)-(2)(u)parallel to parallel to parallel to(1,2)((D)(R-N))(-1), where the parameter region satisfies kappa >= 2, 3 <= N < 6 - mu , mu is an element of ( 0 , N) with 0 < mu <= 4, or in the case of dimension N >= 3 and kappa = 1, mu is an element of ( 0 , N) with 0 < mu <= 4. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies. (AU) | |
| FAPESP's process: | 22/16097-2 - Modern methods in differential geometry and geometric analysis |
| Grantee: | Paolo Piccione |
| Support Opportunities: | Research Projects - Thematic Grants |