2021/06/17 by Juncheng Wei, Wei, Juncheng, Yuanze Wu +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2106.09253
In this paper, we consider the Caffarelli-Kohn-Nirenberg (CKN) inequality: (∫_\mathbb RN|x|-b(p+1)|u|p+1dx)(2)/(p+1)≤ Ca,b,N∫_\mathbb RN|x|-2a|∇ u|2dx where N≥3, a0, where bFS(a) is the Felli-Schneider curve. We prove that in the above parameter region the following stabilities hold: \beginenumerate \item[(1)] stability of CKN inequality in the functional inequality setting dist_D1,2a2(u, Z)\lesssim‖u‖2_D1,2a(\mathbb RN)-Ca,b,N-1‖u‖2_Lp+1(|x|-b(p+1),\mathbb RN) where Z= \ c Wτ| c∈\bbr\backslash\0\, τ>0\; \item[(2)] stability of CKN inequality in the critical point setting (in the class of nonnegative functions) \begineqnarray* distDa1,2(u, Z0ν)\lesssim\\aligned &Γ(u), p>2 or ν=1, &Γ(u)|logΓ(u)|\frac12, p=2 and ν≥2, &Γ(u)(p)/(2), 10\.