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

Abstract

By employing a novel 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 $$ -\left(a+ b\int_{\R^3}|\nabla u|^2\right)\triangle {u}+V(x)u=f(u),\,\,x\in\R^3, $$ where $a,b>0$ are constants, $V\in C(\R^3,\R)$, $f\in C(\R,\R)$. The methodology proposed in the current paper is robust, in the sense that, the monotonicity condition for the nonlinearity $f$ and the coercivity condition of $V$ are not required. Our result improves the study made by Y. Deng, S. Peng and W. Shuai ({\it J. Functional Analysis}, 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 that whether there exist infinitely many sign-changing solutions to the problem above without the coercivity condition of $V$. Moreover, {\it energy doubling} is established, i.e., the energy of sign-changing solutions is strictly large than two times that of the ground state solutions for small $b>0$.