TY - JOUR

T1 - A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity

AU - Liu, Zhisu

AU - Lou, Yijun

AU - Zhang, Jianjun

N1 - Funding Information:
Z. Liu was partially supported by the NSFC (Grant No. 11701267), and Hunan Natural Science Excellent Youth Fund (2020JJ3029), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan, Grant number: CUG2106211; CUGST2). J. Zhang is the corresponding author and was supported by the NSFC (Grant No. 11871123).
Publisher Copyright:
© 2021, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/6

Y1 - 2022/6

N2 - 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.

AB - 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.

KW - Invariant sets of descending flow

KW - Kirchhoff equation

KW - Nonlocal perturbation approach

KW - Sign-changing solution

UR - http://www.scopus.com/inward/record.url?scp=85115098753&partnerID=8YFLogxK

U2 - 10.1007/s10231-021-01155-w

DO - 10.1007/s10231-021-01155-w

M3 - Journal article

AN - SCOPUS:85115098753

VL - 201

SP - 1229

EP - 1255

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

SN - 0373-3114

IS - 3

ER -