Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 61, pp. 1-14.

Title: Ground state solutions for fractional p-Kirchhoff equation
       
Authors: Lixiong Wang (Central South Univ., Changsha, Hunan, China)
         Haibo Chen (Central South Univ., Changsha, Hunan, China)
         Liu Yang (Hengyang Normal Univ., Hengyang, Hunan, China)

Abstract:   
We study the fractional p-Kirchhoff equation
$$
 \Big( a+b  \int_{\mathbb{R}^N}{\int_{\mathbb{R}^N}}
\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dx\, dy\Big)
(-\Delta)_p^s u-\mu|u|^{p-2}u=|u|^{q-2}u, \quad
x\in\mathbb{R}^N,
$$
where $(-\Delta)_p^s$ is the fractional p-Laplacian operator, a and b are strictly
positive real numbers, $s \in (0,1)$, 1 < p< N/s, and p< q< p^*_s-2 with
$p^*_s=\frac{Np}{N-ps}$.   By using the variational method, we prove the existence
and uniqueness of global minimum or mountain pass type critical points on the
$L^p$-normalized manifold
$S(c):=\big\{u\in  W^{s,p}(\mathbb{R}^N): \int_{\mathbb{R}^N} |u|^pdx=c^p\big\}$.

Submitted April 11, 2022. Published August 19, 2022.
Math Subject Classifications: 35J20, 35J60.
Key Words: Variational method; L^p-normalized critical point; fractional; 
           p-Kirchhoff equation; uniqueness.