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

TL;DR

This paper proves the existence and local uniqueness of multi-peak positive solutions for a class of fractional Kirchhoff equations with nonlocal operators and potential functions, using Lyapunov--Schmidt reduction and Pohozăev identities.

Contribution

It introduces a novel approach combining nondegeneracy and reduction methods to handle the coupled nonlocal fractional Kirchhoff problem.

Findings

Existence of multi-peak solutions for small perturbation parameter .

Establishment of local uniqueness of solutions under additional potential conditions.

Handling the interplay between nonlocal operators and Kirchhoff terms in fractional PDEs.

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.