2023/08/11 by Shengbing Deng, Deng, Shengbing, Xingliang Tian +1 · 1 citation
Mathematics · #35J30 #35P30 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2308.06014
openalex publication_date 2023/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is devoted to radial solutions of the following weighted fourth-order equation div(|x|α∇(div(|x|α∇ u)))=u^2**α-1, ugt;0 in ℝN, where N≥ 2, (4-N)/(2)<α<2 and 2**α=(2N)/(N-4+2α). It is obvious that the solutions of above equation are invariant under the scaling λ(N-4+2α)/(2)u(λx) while they are not invariant under translation when α≠ 0. We characterize all the solutions to the related linearized problem about radial solutions, and obtain the conclusion of that if α satisfies (2-α)(2N-2+α)≠4k(N-2+k) for all k∈ℕ+ the radial solution is non-degenerate, otherwise there exist new solutions to the linearized problem that ``replace'' the ones due to the translations invariance. As applications, firstly we investigate the remainder terms of some inequalities related to above equation. Then when N≥ 5 and 0<α<2, we establish a new type second-order Caffarelli-Kohn-Nirenberg inequality ∫ℝN |div(|x|α∇ u)|2 dx ≥ C (∫ℝN|u|^2**α dx)^\frac22**α, for all u∈ C^∞0(ℝN), and in this case we consider a prescribed perturbation problem by using Lyapunov-Schmidt reduction.