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

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.