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

Xiao-Jing Zhong; Chun-Lei Tang

doi:10.3934/cpaa.2017030

Article Contents

2017, Volume 16, Issue 2: 611-628. Doi: 10.3934/cpaa.2017030

This issue Previous Article Existence and stability of periodic solutions for relativistic singular equations Next Article Regularity estimates for continuous solutions of α-convex balance laws

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

Xiao-Jing Zhong^1,2, , and
Chun-Lei Tang^1,2, ,

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
* Corresponding author

Received: June 30, 2016

Revised: August 31, 2016

Published: August 2017

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

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Abstract

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].

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35J65.

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References

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