The existence of normalized solutions to the fractional Kirchhoff equation with potentials

Abstract

In this paper, we delve into the following fractional Kirchhoff equation:$(a+b\underset{R^3}{\int }|{(-\Delta )}^{\frac{s}{2}}u\left|dx\right){(-\Delta )}^{s}u+V\left(x\right)u+\lambda u=|u{|}^{q-2}u+K(x\left)\right|u{|}^{p-2}u,x\in R^3,$ with prescribed mass$\underset{R^3}{\int }|u{|}^{2}dx=c^2,$ where $s\in (\frac{3}{4},1)$, $a,b,c>0$, $p\in [1,2)$, $\lambda \in R$, $V\left(x\right)\not\equiv 0,K\left(x\right)\not\equiv 0$. This paper focuses on two cases. Firstly, under specific assumptions where the potentials satisfy $V\left(x\right)\ge 0$ and $K\left(x\right)\le 0$, we employ the linking geometry method to rigorously prove the existence of at least one $L^2$-normalized solution $(u,\lambda )\in H^s\left(R^3\right)\times R^{+}$ to the equation. Secondly, shifting our focus to scenarios where the potentials adhere to $V\left(x\right)\le 0$ and $K\left(x\right)\ge 0$, we demonstrate the existence of a mountain pass $L^2$-normalized solution with positive energy.