Infinitely many sign-changing solutions for Kirchhoff type problems in ${\mathbb{R}}^3$

Abstract

In this paper, we consider the following nonlinear Kirchhoff type problem: \[ \left\{\begin{array}{lcl}-\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2\right)Δu+V(x)u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3), \end{array}\right. \] where $a,b>0$ are constants, the nonlinearity $f$ is superlinear at infinity with subcritical growth and $V$ is continuous and coercive. For the case when $f$ is odd in $u$ we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity $|u{|}^{p-2}u$ with $p\in (2,4]$.