A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity

Zhisu Liu, Yijun Lou, Jianjun Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

6 Citations (Scopus)

Abstract

By employing a nonlocal perturbation approach and the method of invariant sets of descending flow, this manuscript investigates the existence and multiplicity of sign-changing solutions to a class of semilinear Kirchhoff equations in the following form -(a+b∫R3|∇u|2)Δu+V(x)u=f(u),x∈R3,where a, b> 0 are constants, V∈ C(R3, R) , f∈ C(R, R). The methodology proposed in the current paper is robust, in the sense that, neither the monotonicity condition on f nor the coercivity condition on V is required. Our result improves the study made by Deng et al. (J Funct Anal 269:3500–3527, 2015), in the sense that, in the present paper, the nonlinearities include the power-type case f(u) = | u| p-2u for p∈ (2 , 4) , in which case, it remains open in the existing literature whether there exist infinitely many sign-changing solutions to the problem above. Moreover, energy doubling is established, namely, the energy of sign-changing solutions is strictly larger than two times that of the ground state solutions for small b> 0.

Original languageEnglish
Pages (from-to)1229-1255
Number of pages27
JournalAnnali di Matematica Pura ed Applicata
Volume201
Issue number3
DOIs
Publication statusPublished - Jun 2022

Keywords

  • Invariant sets of descending flow
  • Kirchhoff equation
  • Nonlocal perturbation approach
  • Sign-changing solution

ASJC Scopus subject areas

  • Applied Mathematics

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