Infinitely Many Bubbling Solutions and Non-Degeneracy Results to Fractional Prescribed Curvature Problems

We consider the following fractional prescribed curvature problem

$$(-\Delta)^{s}u=K(y)u^{2_{s}^{*}-1}, \quad u>0,\,y \in {\mathbb R}^{N},$$

((0.1))

where \(s \in (0,\, {1 \over 2})\) for N = 3, s ∈ (0, 1) for N ≥ 4 and \(2_{s}^{*}={2N \over N-2s}\) 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 concentrates 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.