The existence of sign-changing solutions for Schrödinger-Kirchhoff problems in R³

Abstract

In this paper, we consider the following Kirchhoff-type equation: $ -\left(a+b\int_{ \mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+u = |u|^{p-1}u,\quad {\rm{in }}\; \mathbb{R}^3, $ where $ a $, $ b > 0 $, $ p \in (1, 5) $. By considering a minimization problem on a special constraint set, we prove that the above problem has at least one sign-changing solution for any $ p \in (1, 5) $. Our results (especially $ p \in (1, 3] $) can be regarded as an improvement on the existing results.

Keywords

• kirchhoff type equation
• sign-changing solution
• constraint variational method
• pohožaev identity