Acta mathematica scientia,Series B ›› 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
  • 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).

Abstract: For any s(0,1), let the nonlocal Sobolev space Xs(RN) be the linear space of Lebesgue measure functions from RN to R such that any function u in Xs(RN) belongs to L2(RN) and the function (x,y)(u(x)u(y))K(xy) is in L2(RN,RN). First, we show, for a coercive function V(x), the subspace E:={uXs(RN):RNV(x)u2dx<+} of Xs(RN) is embedded compactly into Lp(RN) for p[2,2s), where 2s 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 LKu+V(x)u=f(x,u), x RN are obtained, where LK is an integro-differential operator and V is coercive at infinity.

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

CLC Number: 

  • 35R11
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