Infinitely many bubbling solutions and non-degeneracy results to fractional prescribed curvature problems

2022/07/29 by Lixiu Duan, Qing Guo, Duan, Lixiu +1
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2207.14441

openalex publication_date 2022/07/29 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28

Abstract

We consider the following fractional prescribed curvature problem (-Δ)^s u= K(y)u^{2^*_s-1}, ugt;0, y∈ ℝ^N, (0.1) where s∈(0,(1)/(2)) for N=3, s∈(0,1) for N\geqslant4 and 2^*_s=(2N)/(N-2s) is the fractional critical Sobolev exponent, K(y) has a local maximum point in r∈(r₀-δ,r₀+δ). First, for any sufficient large k, we construct a 2k bubbling solution to (0.1) of some new type, which concentrate on an upper and lower surfaces of an oblate cylinder through the Lyapunov-Schmidt reduction method. Furthermore, a non-degeneracy result of the multi-bubbling solutions is proved by use of various Pohozaev identities, which is new in the study of the fractional problems.

Related

• New type of bubbling solutions to a critical fractional Schrödinger equation with double potentials
• Sign-changing bubble-tower solutions to fractional semilinear elliptic\n problems
• Peak Solutions for the fractional Nirenberg problem
• Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay
• Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian