# American Institute of Mathematical Sciences

March  2017, 16(2): 611-628. doi: 10.3934/cpaa.2017030

## The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem

 1 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China 2 School of History Culture and Ethnology, Southwest University, Chongqing 400715, China

* Corresponding author

Received  July 2016 Revised  September 2016 Published  January 2017

Fund Project: supported by National Natural Science Foundation of China(No. 11471267); the Fundamental Research Funds for the Central Universities (No. SWU1109075).

In this paper, we investigate the existence and nonexistence of ground state nodal solutions to a class of Kirchhoff type problems
 $-\left( a+b\int_{\Omega }{|}\nabla u{{|}.{2}}dx \right)\vartriangle u=\lambda u+|u{{|}.{2}}u,\ \ u\in H_{0}.{1}(\Omega ),$
where $a, b>0$, $\lambda < a\lambda_1$, $\lambda_1$ is the principal eigenvalue of $(-\triangle, H_0.{1}(\Omega))$. With the help of the Nehari manifold, we obtain that there is $\Lambda>0$ such that the Kirchhoff type problem possesses at least one ground state nodal solution $u_b$ for all $0 < b < \Lambda$ and $\lambda < a\lambda_1$ and prove that its energy is strictly larger than twice that of ground state solutions. Moreover, we give a convergence property of $u_b$ as $b\searrow 0$. Besides, we firstly establish the nonexistence result of nodal solutions for all $b\geq\Lambda$. This paper can be regarded as the extension and complementary work of W. Shuai (2015)[21], X.H. Tang and B.T. Cheng (2016)[22].
Citation: Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure & Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030
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