Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type

Wen  Zhang; Xianhua Tang; Bitao  Cheng; Jian  Zhang

doi:10.3934/cpaa.2016032

Article Contents

2016, Volume 15, Issue 6: 2161-2177. Doi: 10.3934/cpaa.2016032

This issue Previous Article Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates Next Article Sharp well-posedness for the Chen-Lee equation

Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type

1.
School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan
2.
School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083
3.
School of Mathematics and Statistics, Qujing Normal University, Qujing, 655011 Yunnan

Received: November 30, 2015

Revised: February 29, 2016

Published: October 2016

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Abstract

In this paper, we study the following fourth-order elliptic equation with Kirchhoff-type \begin{eqnarray} \left\{\begin{array}{l} \Delta^{2}u-(a+b\int_{\mathbb{R}^N}|\nabla u|^{2}dx)\Delta u+V(x)u=f(u),\ \ \ x\in \mathbb{R}^{N},\\ u\in H^{2}(\mathbb{R}^{N}), \end{array}\right. \end{eqnarray} where the constants $a>0, b\geq 0$. By constraint variational method and quantitative deformation lemma, we obtain that the problem possesses one least energy sign-changing solution $u_{b}$. Moreover, we also prove that the energy of $u_{b}$ is strictly larger than two times the ground state energy. Finally, we give a convergence property of $u_{b}$ when $b$ as a parameter and $b\rightarrow 0$.

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Mathematics Subject Classification: 35J50, 35J60.

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