Electron. J. Differential Equations, Vol. 2018 (2018), No. 144, pp. 1-19.

Existence of multiple solutions and estimates of extremal values for a Kirchhoff type problem with fast increasing weight and critical nonlinearity

Xiaotao Qian, Jianqing Chen

Abstract:
In this article, we study the Kirchhoff type problem
$$ -\Big(a+\epsilon\int_{\mathbb{R}^3} K(x)|\nabla u|^2dx\Big)\hbox{div} (K(x)\nabla u)=\lambda K(x)f(x)|u|^{q-2}u+K(x)|u|^{4}u, $$
where $x\in \mathbb{R}^3$, $1<q<2$, $K(x)=\exp({|x|^{\alpha}/4})$ with $\alpha\geq2$, $\epsilon>0$ is small enough, and the parameters $a, \lambda >0$. Under some assumptions on $f(x)$, we establish the existence of two nonnegative nontrivial solutions and obtain uniform lower estimates for extremal values of the problem via variational methods.

Submitted February 10, 2018. Published July 17, 2018.
Math Subject Classifications: 35J20, 35J60.
Key Words: Variational methods; Kirchhoff type equation; critical nonlinearity; multiple solutions; extremal values.

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Xiaotao Qian
College of Mathematics and Computer Science
& FJKLMAA, Fujian Normal University
Qishan Campus, Fuzhou 350108, China
email: qianxiaotao1984@163.com
Jianqing Chen
College of Mathematics and Computer Science
& FJKLMAA, Fujian Normal University
Qishan Campus, Fuzhou 350108, China
email: jqchen@fjnu.edu.cn

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