Least energy sign-changing solutions of Kirchhoff equation on bounded domains

Abstract

We deal with sign-changing solutions for the Kirchhoff equation $ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ u = 0, \; \ x\in \partial\Omega, \end{cases} $ where $ a, b > 0 $ and $ \lambda, \mu\in\mathbb{R} $ being parameters, $ \Omega\subset \mathbb{R}^{3} $ is a bounded domain with smooth boundary $ \partial\Omega $. Combining Nehari manifold method with the quantitative deformation lemma, we prove that there exists $ \mu^{\ast} > 0 $ such that above problem has at least a least energy sign-changing (or nodal) solution if $ \lambda < a\lambda_{1} $ and $ \mu > \mu^{\ast} $, where $ \lambda_{1} > 0 $ is the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. It is noticed that the nonlinearity $ \lambda u+\mu|u|^{2}u $ fails to satisfy super-linear near zero and super-three-linear near infinity, respectively.

Keywords

• kirchhoff equation
• nonlocal term
• variation methods
• sign-changing solutions