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

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

$$\begin{aligned} -\left( a+ b\int _{\mathbb {R}^3}|\nabla u|^2\right) \Delta {u}+V(x)u=f(u),\,\,x\in \mathbb {R}^3, \end{aligned}$$

where $a,b>0$ are constants, $V\in C(\mathbb {R}^3,\mathbb {R})$, $f\in C(\mathbb {R},\mathbb {R})$. 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 $f(u)=|u|^{p-2}u$ for $p\in (2,4)$, 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 $b>0$.