In this paper we discuss the following Kirchhoff equation

\left\{
\begin{array}{lr}
-\left(a+b \int_{\mathbb{R}^3}|\nabla u|^{2} d x\right) \Delta u+V(x)u+\lambda u=\mu|u|^{q-2}u+|u|^{p-2}u \ {\rm in}\ \mathbb{R}^3,&\\
\int_{\mathbb{R}^{3}}u^{2}dx=c^2,
\end{array}
\right.
where a, b, µ and c are positive numbers, λ is unknown and appears as a Lagrange multiplier,

14/3<q<p<6 and V is a continuous non-positive function vanishing at infinity.
Under some mild assumptions on V , we prove the existence of a mountain pass normalized solution. To the author’s knowledge, it is the first time to study the existence of
normalized solution to Kirchhoff equation with potential via the minimax principle.

 In this paper we discuss the following Kirchhoff equation

\left\{
\begin{array}{lr}
-\left(a+b \int_{\mathbb{R}^3}|\nabla u|^{2} d x\right) \Delta u+V(x)u+\lambda u=\mu|u|^{q-2}u+|u|^{p-2}u \ {\rm in}\ \mathbb{R}^3,&\\
\int_{\mathbb{R}^{3}}u^{2}dx=c^2,
\end{array}
\right.
where a, b, µ and c are positive numbers, λ is unknown and appears as a Lagrange multiplier,

14/3<q<p<6 and V is a continuous non-positive function vanishing at infinity.
Under some mild assumptions on V , we prove the existence of a mountain pass normalized solution. To the author’s knowledge, it is the first time to study the existence of
normalized solution to Kirchhoff equation with potential via the minimax principle.