Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence, non-degeneracy and local uniqueness of multi-peak solutions to the fractional Schrödinger equation with nearly critical exponent in $\mathbb{R}^N$

Yanyan Guo, Ying Li, Zhongyuan Liu, Pingping Yang

Abstract

In this paper, we consider the following fractional Schrödinger equation \begin{equation*} \left\{ \begin{array}{lcl} (-Δ)^{s}u+V(x)u=u^{{p_s}-ε}\ \ \ &\hbox{in}\ \mathbb{R}^N,\\ u>0\ \ \ &\hbox{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where $0<s<1$, $ε>0$, $p_s=(N+2s)/(N-2s)$, $N>4s$ and $V(x)\in C^1(\mathbb{R}^N)\cap L^\infty (\mathbb{R}^N)$ is non-negative. We first use the Lyapunov-Schmidt reduction method to construct multi-peak solutions to the above equation provided that $V(x)$ possesses $k$ stable critical points. Then we prove the non-degeneracy and local uniqueness of the multi-peak solutions, for $\frac{1}{2}<s<1$, $N\geq 6s$, via the blow-up argument based on various local Pohozaev identities. Due to the nonlocal property of the fractional Laplacian, we need to make delicate analysis of the approximate solutions and establish the local Pohozaev identities for the corresponding harmonic extension instead of $u$. This approach not only requires to develop refined estimates for several integrals in the local Pohozaev identities, but also to apply Pohozaev identities through a markedly different way.

Existence, non-degeneracy and local uniqueness of multi-peak solutions to the fractional Schrödinger equation with nearly critical exponent in $\mathbb{R}^N$

Abstract

In this paper, we consider the following fractional Schrödinger equation \begin{equation*} \left\{ \begin{array}{lcl} (-Δ)^{s}u+V(x)u=u^{{p_s}-ε}\ \ \ &\hbox{in}\ \mathbb{R}^N,\\ u>0\ \ \ &\hbox{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where , , , and is non-negative. We first use the Lyapunov-Schmidt reduction method to construct multi-peak solutions to the above equation provided that possesses stable critical points. Then we prove the non-degeneracy and local uniqueness of the multi-peak solutions, for , , via the blow-up argument based on various local Pohozaev identities. Due to the nonlocal property of the fractional Laplacian, we need to make delicate analysis of the approximate solutions and establish the local Pohozaev identities for the corresponding harmonic extension instead of . This approach not only requires to develop refined estimates for several integrals in the local Pohozaev identities, but also to apply Pohozaev identities through a markedly different way.
Paper Structure (18 sections, 42 theorems, 319 equations)

This paper contains 18 sections, 42 theorems, 319 equations.

Table of Contents

  1. Introduction and main results
  2. Motivation and known results
  3. Problem setup and main results
  4. The strategy of the proof
  5. Preliminaries
  6. The Lyapunov–Schmidt reduction
  7. Proof of the Main Theorem
  8. Proof of Theorem \ref{['NDth']}
  9. Basic Estimates and Decomposition of $\omega_\epsilon$
  10. Pohozaev Identities and Coefficients Estimates
  11. Completion of the Proof of Theorem \ref{['NDth']}
  12. Proof of Theorem \ref{['thm1.3']}
  13. Pohozaev identities and refined estimates
  14. Lyapunov-Schmidt decomposition
  15. Estimate of $d_i^l$, $l=0,1,\cdots,N$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $N>4s$ and $0<s<1$. Assume that $V(x)$ is a bounded nonnegative $C^1(\mathbb{R}^N)$ function and $\xi^*_i$, $i=1,2,\cdots,k$ are the $k$ different stable critical points of $V(x)$ with $V(\xi^*_i)>0$. Then there exists a constant $\epsilon_0>0$ such that for $\epsilon\in (0,\epsilon_0)$, equatio where $\bm\lambda_\epsilon=(\lambda_1^\epsilon,\cdots,\lambda_k^\epsilon)$, $\bm\xi_\epsilon=(\xi_1

Theorems & Definitions (80)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Theorem 1.2
  • Theorem 1.3
  • Remark 3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 70 more