Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations

Abstract

This paper has two main purposes. In the first part, combining the nondegeneracy of the ground state with the Lyapunov--Schmidt reduction method, we prove the existence of multi-peak positive solutions to the singularly perturbed problem \[\Big(\varepsilon^{2s}a+\varepsilon^{4s-N} b\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^2\,dx\Big)(-\Delta)^s u+V(x)u=u^p\quad \text{in }\mathbb{R}^{N},\] for all sufficiently small $\varepsilon> 0$, under the assumptions $2s<N<4s$, $1<p<2^*_s-1$, and some mild conditions on the potential $V$. The main difficulty comes from the interplay between the nonlocal operator $(-\Delta)^s$ and the nonlocal Kirchhoff term, which makes the corresponding limiting problem a coupled system of partial differential equations rather than a single fractional Kirchhoff equation. In the second part, under additional assumptions on $V$, we establish the local uniqueness of positive multi-peak solutions by means of a local Pohoz\v{a}ev identity.