数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (5): 1853-1876.doi: 10.1007/s10473-024-0512-5

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A COMPACT EMBEDDING RESULT FOR NONLOCAL SOBOLEV SPACES AND MULTIPLICITY OF SIGN-CHANGING SOLUTIONS FOR NONLOCAL SCHRÖDINGER EQUATIONS*

Xu Zhang1,2, Hao zhai3, Fukun zhao3,†   

  1. 1. Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China;
    2. Department of Mathematics, Yunnan Normal University, Kunming 650500, China;
    3. Department of Mathematics, Yunnan Normal University, Kunming 650500, China
  • 收稿日期:2023-01-13 修回日期:2024-05-04 出版日期:2024-10-25 发布日期:2024-10-22
  • 通讯作者: †Fukun zhao , E-mail,: fukunzhao@163.com
  • 作者简介:Xu Zhang, E-mail,: zhangxu0606@163.com; Hao ZHAI,E-mail,: 905358972@qq.com
  • 基金资助:
    NSFC (12261107) and Yunnan Key Laboratory of Modern Analytical Mathematics and Applications (202302AN360007).

A COMPACT EMBEDDING RESULT FOR NONLOCAL SOBOLEV SPACES AND MULTIPLICITY OF SIGN-CHANGING SOLUTIONS FOR NONLOCAL SCHRÖDINGER EQUATIONS*

Xu Zhang1,2, Hao zhai3, Fukun zhao3,†   

  1. 1. Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China;
    2. Department of Mathematics, Yunnan Normal University, Kunming 650500, China;
    3. Department of Mathematics, Yunnan Normal University, Kunming 650500, China
  • Received:2023-01-13 Revised:2024-05-04 Online:2024-10-25 Published:2024-10-22
  • Contact: †Fukun zhao , E-mail,: fukunzhao@163.com
  • About author:Xu Zhang, E-mail,: zhangxu0606@163.com; Hao ZHAI,E-mail,: 905358972@qq.com
  • Supported by:
    NSFC (12261107) and Yunnan Key Laboratory of Modern Analytical Mathematics and Applications (202302AN360007).

摘要: For any $s\in(0,1)$, let the nonlocal Sobolev space $X^s(\mathbb{R} ^N)$ be the linear space of Lebesgue measure functions from $\mathbb{R} ^N$ to $\mathbb{R} $ such that any function $u$ in $X^s(\mathbb{R} ^N)$ belongs to $L^2(\mathbb{R} ^N)$ and the function $(x,y)\longmapsto\big(u(x)-u(y)\big)\sqrt{K(x-y)}$ is in $L^2(\mathbb{R} ^N,\mathbb{R} ^N)$. First, we show, for a coercive function $V(x)$, the subspace $E:=\bigg\{u\in X^s(\mathbb{R} ^N):\int_{\mathbb{R} ^N}V(x)u^2{\rm d}x<+\infty\bigg\}$ of $X^s(\mathbb{R} ^N)$ is embedded compactly into $L^p(\mathbb{R}^N)$ for $p\in[2,2_s^*)$, where $2_s^*$ is the fractional Sobolev critical exponent. In terms of applications, the existence of a least energy sign-changing solution and infinitely many sign-changing solutions of the nonlocal Schrödinger equation $-{\mathcal{L}_K}u+V(x)u=f(x,u),\ x\in\ \mathbb{R} ^N$ are obtained, where $-{\mathcal{L}_K}$ is an integro-differential operator and $V$ is coercive at infinity.

关键词: sign-changing solution, integro-differential operator, least energy, variational method

Abstract: For any $s\in(0,1)$, let the nonlocal Sobolev space $X^s(\mathbb{R} ^N)$ be the linear space of Lebesgue measure functions from $\mathbb{R} ^N$ to $\mathbb{R} $ such that any function $u$ in $X^s(\mathbb{R} ^N)$ belongs to $L^2(\mathbb{R} ^N)$ and the function $(x,y)\longmapsto\big(u(x)-u(y)\big)\sqrt{K(x-y)}$ is in $L^2(\mathbb{R} ^N,\mathbb{R} ^N)$. First, we show, for a coercive function $V(x)$, the subspace $E:=\bigg\{u\in X^s(\mathbb{R} ^N):\int_{\mathbb{R} ^N}V(x)u^2{\rm d}x<+\infty\bigg\}$ of $X^s(\mathbb{R} ^N)$ is embedded compactly into $L^p(\mathbb{R}^N)$ for $p\in[2,2_s^*)$, where $2_s^*$ is the fractional Sobolev critical exponent. In terms of applications, the existence of a least energy sign-changing solution and infinitely many sign-changing solutions of the nonlocal Schrödinger equation $-{\mathcal{L}_K}u+V(x)u=f(x,u),\ x\in\ \mathbb{R} ^N$ are obtained, where $-{\mathcal{L}_K}$ is an integro-differential operator and $V$ is coercive at infinity.

Key words: sign-changing solution, integro-differential operator, least energy, variational method

中图分类号: 

  • 35R11