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

Abstract

The paper is concerned with the p-Kirchhoff equation

$$\begin{aligned} -\left( a+b\int _{\mathbb {R}^{N}}|\nabla u|^{p}dx\right) \Delta _{p} u=f(u)-\mu u-V(x)u^{p-1}~~~~~in~~H^{1}(\mathbb {R}^{N}), \end{aligned}$$

(1)

where \(a,b>0\). When \(V(x)=0\), \(p=2\) and \(N\ge 3\), 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 \(\mathbb {R}^{N}\) by using some energy estimates. When \(V(x)\not \equiv 0, p>\sqrt{3}+1, \frac{2}{p-2}<p\le N<2p\), under an explicit smallness assumption on V with \(\lim _{|x|\rightarrow \infty }V(x)=\sup _{\mathbb {R}^{N}}V(x)\), we prove the existence of energy ground state normalized solutions of (1).