Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity

Chungen Liu; Huabo Zhang; Chungen Liu; Huabo Zhang

doi:10.3934/era.2021038

Electronic Research Archive

2021, Volume 29, Issue 5: 3281-3295. doi: 10.3934/era.2021038

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Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity

Chungen Liu ^,,
Huabo Zhang

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

Received: 01 March 2021 Revised: 01 April 2021 Published: 26 May 2021
Primary: 58F15, 58F17; Secondary: 53C35

In this paper, we consider the existence of least energy nodal solution and ground state solution, energy doubling property for the following fractional critical problem
$ \begin{cases} -(a+ b\|u\|_{K}^{2})\mathcal{L}_K u+V(x)u = |u|^{2^{\ast}_{\alpha}-2}u+ k f(x,u),&x\in\Omega,\\ u = 0,&x\in\mathbb{R}^{3}\backslash\Omega, \end{cases} $
where $ k $ is a positive parameter, $ \mathcal{L}_K $ stands for a nonlocal fractional operator which is defined with the kernel function $ K $. By using the nodal Nehari manifold method, we obtain a least energy nodal solution $ u $ and a ground state solution $ v $ to this problem when $ k\gg1 $, where the nonlinear function $ f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow \mathbb{R} $ is a Carathéodory function.
- Fractional Kirchhoff equation,
- nodal solution,
- ground state solution,
- Nehari manifold
Citation: Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity[J]. Electronic Research Archive, 2021, 29(5): 3281-3295. doi: 10.3934/era.2021038

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Abstract

In this paper, we consider the existence of least energy nodal solution and ground state solution, energy doubling property for the following fractional critical problem

$ \begin{cases} -(a+ b\|u\|_{K}^{2})\mathcal{L}_K u+V(x)u = |u|^{2^{\ast}_{\alpha}-2}u+ k f(x,u),&x\in\Omega,\\ u = 0,&x\in\mathbb{R}^{3}\backslash\Omega, \end{cases} $

where $ k $ is a positive parameter, $ \mathcal{L}_K $ stands for a nonlocal fractional operator which is defined with the kernel function $ K $. By using the nodal Nehari manifold method, we obtain a least energy nodal solution $ u $ and a ground state solution $ v $ to this problem when $ k\gg1 $, where the nonlinear function $ f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow \mathbb{R} $ is a Carathéodory function.

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