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 $\left\{\begin{array}{l} \left(a+b \int_{\mathbb{R}^{3}}\left[|\nabla u|^{2}+V(x) u^{2}\right] d x\right)[-\Delta u+V(x) u]=\lambda u+\mu|u|^{q-2} u+|u|^{p-2} u, \quad \text { in } \mathbb{R}^{3} \\ \int_{\mathbb{R}^{3}}|u|^{2} d x=c^{2} \end{array}\right.$ where $a>0, b>0, \lambda \in \mathbb{R}, 5<q<p<6, \mu>0$ 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.

Keywords

Kirchhoff equation / normalized solutions / minimax principle