### Electronic Research Archive

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

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

• 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.

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

### Related Papers:

• 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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