Normalized Solutions to the Kirchhoff Equation with Potential Term: Mass Super-Critical Case

Abstract

In this paper, we study the existence of normalized solution to the following nonlinear mass super-critical Kirchhoff equation

$$\begin{aligned} \left\{ \begin{aligned}&-\left( a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\right) \triangle u+V(x)u+\lambda u=g(u) \ \ {in} \ {{\mathbb {R}}^{N}}\\&0\le u\in H^{1}_{r}({\mathbb {R}}^{N}) \end{aligned}\right. \end{aligned}$$

where $a ,b>0$ are constants, $\lambda \in R$, and V(x) satisfies appropriate assumptions; g has a mass super-critical growth when $N=3$, and $g(u)=|u|^{p-2}u$ with $p\in (2+\frac{8}{N},2^{*}), 2^{*}=\frac{2N}{N-2}$ when $N\ge 3$. Here, we prove the existence of ground state normalized solution via variational methods.