Normalized Solutions for a Kirchhoff Equation with Potential in $\mathbb{R}^3$

Abstract

In this paper, for given mass $c>0,$ we study the existence of normalized solutions to the following nonlinear Kirchhoff equation  $$\begin{cases} (a+b\int_{\mathbb{R}^3}[|\nabla u|^2+V(x)u^2]dx)[-\Delta u+V(x)u]=\lambda u+\mu|u|^{q-2}u+|u|^{p-2}u, \ \ \ {\rm in}\ \ \mathbb{R}^3, \\ \int_{\mathbb{R}^3}|u|^2dx=c^2, \end{cases}$$where $a>0, b>0, λ∈\mathbb{R},$ $50$ 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 via the minimax principle.