On general Kirchhoff type equations with steep potential well and critical growth in $ \mathbb{R}^2 $

Zhenluo Lou; Jian Zhang; Zhenluo Lou; Jian Zhang

doi:10.3934/math.20241041

AIMS Mathematics

2024, Volume 9, Issue 8: 21433-21454. doi: 10.3934/math.20241041

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On general Kirchhoff type equations with steep potential well and critical growth in $ \mathbb{R}^2 $

Zhenluo Lou ^{1
,
,},
Jian Zhang ²

1.
School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, Henan, China
2.
College of Science, China University of Petroleum, Qingdao 266580, Shandong, China

Received: 27 March 2024 Revised: 13 June 2024 Accepted: 20 June 2024 Published: 04 July 2024
MSC : 35A01, 35A15, 35J20

In this paper, we study the following Kirchhoff-type equation:$ \begin{equation*} M\left(\displaystyle{\int}_{\mathbb{R}^2}(|\nabla u|^2 +u^2)\mathrm{d} x\right)(-\Delta u+u) + \mu V(x)u = K(x) f(u) \ \ \mathrm{in} \ \ \mathbb{R}^2, \end{equation*} $where $ M \in C(\mathbb{R}^+, \mathbb{R}^+) $ is a general function, $ V \geq 0 $ and its zero set may have several disjoint connected components, $ \mu > 0 $ is a parameter, $ K $ is permitted to be unbounded above, and $ f $ has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhoff-type equations with critical growth in the whole space, we get the existence and concentration behavior of solutions. The results are new even for the case $ M \equiv 1 $.
- Kirchhoff-type equation,
- steep potential well,
- critical growth,
- dimension two,
- variational method
Citation: Zhenluo Lou, Jian Zhang. On general Kirchhoff type equations with steep potential well and critical growth in $ \mathbb{R}^2 $[J]. AIMS Mathematics, 2024, 9(8): 21433-21454. doi: 10.3934/math.20241041

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Abstract

In this paper, we study the following Kirchhoff-type equation:$ \begin{equation*} M\left(\displaystyle{\int}_{\mathbb{R}^2}(|\nabla u|^2 +u^2)\mathrm{d} x\right)(-\Delta u+u) + \mu V(x)u = K(x) f(u) \ \ \mathrm{in} \ \ \mathbb{R}^2, \end{equation*} $where $ M \in C(\mathbb{R}^+, \mathbb{R}^+) $ is a general function, $ V \geq 0 $ and its zero set may have several disjoint connected components, $ \mu > 0 $ is a parameter, $ K $ is permitted to be unbounded above, and $ f $ has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhoff-type equations with critical growth in the whole space, we get the existence and concentration behavior of solutions. The results are new even for the case $ M \equiv 1 $.

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