Normalized solution to p-Kirchhoff-type equation in \(\mathbb {R}^{N}\)

The paper is concerned with the p-Kirchhoff equation 1 \(\begin{aligned} -\left( a+b\int _{\mathbb {R}^{N}}|\nabla u|^{p}dx\right) \Delta _{p} u=f(u)-\mu u-V(x)u^{p-1}~~~~~in~~H^{1}(\mathbb {R}^{N}), \end{aligned}\) $$\begin{array}{c}\hfill -\left(a,+,b,{\int }_{{\mathbb{R}}^N},{|\mathrm{\nabla }u|}^p,d,x\right){\mathrm{\Delta }}_{p}u=f(u)-\mu u-V(x)u^{p-1}in{H}^1({\mathbb{R}}^N),\end{array}$$ where \(a,b>0\) $a,b>0$ . When \(V(x)=0\) $V(x)=0$ , \(p=2\) $p=2$ and \(N\ge 3\) $N\ge 3$ , we obtain that any energy ground state normalized solutions of (1) has constant sign and is radially symmetric monotone with respect to some point in \(\mathbb {R}^{N}\) ${\mathbb{R}}^N$ by using some energy estimates. When \(V(x)\not \equiv 0, p>\sqrt{3}+1, \frac{2}{p-2}<p\le N<2p\) $V(x)\not\equiv 0,p>\sqrt{3}+1,\frac{2}{p-2}<p\le N<2p$ , under an explicit smallness assumption on V with \(\lim _{|x|\rightarrow \infty }V(x)=\sup _{\mathbb {R}^{N}}V(x)\) ${lim}_{|x|\to \infty }V(x)={sup}_{{\mathbb{R}}^N}V(x)$ , we prove the existence of energy ground state normalized solutions of (1).