Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension

Qilin Xie; Jianshe Yu

doi:10.3934/cpaa.2019008

Article Contents

2019, Volume 18, Issue 1: 129-158. Doi: 10.3934/cpaa.2019008

This issue Previous Article Boundary regularity for a degenerate elliptic equation with mixed boundary conditions Next Article Existence and a general decay results for a viscoelastic plate equation with a logarithmic nonlinearity

Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension

Qilin Xie and
Jianshe Yu^,

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, Guangdong, China

* Corresponding author
* Corresponding author

Received: August 31, 2017

Revised: September 30, 2017

Published: August 2018

The first author is supported by National Natural Science Foundation of China grant 11701113 and China Postdoctoral Science Foundation funded project grant 2016M600647. The second author is supported by National Natural Science Foundation of China grant 11471085, Program for Changjiang Scholars and Innovative Research Team in University grant IRT1226 and Guangdong Innovative Research Team Program grant 2011S009.

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Abstract

In the present paper, we consider the following Kirchhoff type problem

$\begin{cases}-\Big(a+λ∈t_{\mathbb R^N} | \nabla u|^2dx\Big) Δ u+V(x)u = |u|^{2^*-2}u \;\;\;{\rm in}\ \mathbb{R}^N,\\u∈ D^{1,2}(\mathbb R^N),\end{cases}$

where $a$ is a positive constant, $λ$ is a positive parameter, $V∈ L^{\frac{N}{2}}(\mathbb{R}^N)$ is a given nonnegative function and $2^*$ is the critical exponent. The existence of bounded state solutions for Kirchhoff type problem with critical exponents in the whole $\mathbb R^N$ ($N≥5$ ) has never been considered so far. We obtain sufficient conditions on the existence of bounded state solutions in high dimension $N≥4$ , and especially it is the fist time to consider the case when $N≥5$ in the literature.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 47J30, 35J20.

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Figure 1. N=3

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Figure 2. N=4

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Figure 3. $N\geq5$

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