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

Zhipeng Yang

TL;DR

The paper analyzes a singularly perturbed fractional Kirchhoff equation and proves the existence of multi-peak positive solutions concentrating near prescribed points where the potential attains favorable values. A Lyapunov–Schmidt reduction is used to reduce the infinite-dimensional problem to a finite-dimensional one, yielding a coupled system as the limiting equation for the peak profiles. Under mild conditions on the potential, the authors show both existence and precise concentration behavior; with additional assumptions, they establish local uniqueness of these multi-peak states via a local Pohozaev identity. This work reveals a novel phenomenon where the limiting behavior of multi-peak solutions is governed by a coupled system rather than a single fractional Kirchhoff equation, highlighting the nonlocal nature of the Kirchhoff term in the fractional setting.

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 \[\Big(\varepsilon^{2s}a+\varepsilon^{4s-N} b\int_{\mathbb{R}^{N}}|(-Δ)^{\frac{s}{2}}u|^2\,dx\Big)(-Δ)^s u+V(x)u=u^p\quad \text{in }\mathbb{R}^{N},\] 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 $(-Δ)^s$ 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 Pohozǎev identity.

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

TL;DR

The paper analyzes a singularly perturbed fractional Kirchhoff equation and proves the existence of multi-peak positive solutions concentrating near prescribed points where the potential attains favorable values. A Lyapunov–Schmidt reduction is used to reduce the infinite-dimensional problem to a finite-dimensional one, yielding a coupled system as the limiting equation for the peak profiles. Under mild conditions on the potential, the authors show both existence and precise concentration behavior; with additional assumptions, they establish local uniqueness of these multi-peak states via a local Pohozaev identity. This work reveals a novel phenomenon where the limiting behavior of multi-peak solutions is governed by a coupled system rather than a single fractional Kirchhoff equation, highlighting the nonlocal nature of the Kirchhoff term in the fractional setting.

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 for all sufficiently small , under the assumptions , , and some mild conditions on the potential . The main difficulty comes from the interplay between the nonlocal operator 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 , we establish the local uniqueness of positive multi-peak solutions by means of a local Pohozǎev identity.
Paper Structure (7 sections, 20 theorems, 460 equations)

This paper contains 7 sections, 20 theorems, 460 equations.

Table of Contents

  1. Introduction and main results
  2. The form and locations of multi-peak solutions
  3. Some preliminaries
  4. Semiclassical solutions for the fractional Kirchhoff equation
  5. Finite dimensional reduction
  6. Proof of Theorem \ref{['Thm1.1']}
  7. Uniqueness of semiclassical bounded states

Key Result

Proposition 1.1

Let $a,b>0$. Assume that $\frac{N}{4}<s<1$ and $1<p<2_s^*-1$. Then equation eq1.5 has a ground state solution $U\in H^s(\mathbb{R}^N)$ which is unique up to translations, and: Moreover, $U$ is nondegenerate in $H^s(\mathbb{R}^N)$ in the sense that where $L_+$ is defined by acting on $L^2(\mathbb{R}^N)$ with domain $H^s(\mathbb{R}^N)$.

Theorems & Definitions (21)

  • Definition 1.1
  • Proposition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 3.2
  • ...and 11 more