Multiple positive solutions to Kirchhoff equations with competing potential functions in R3 $\mathbb{R}^{3}$

Abstract In this paper, we study the existence of multiple positive solutions to the following Kirchhoff equation with competing potential functions: {−(ε2a+εb∫R3|∇v|2)Δv+V(x)v=K(x)|v|p−1vin R3,v>0,v∈H1(R3), $$ \textstyle\begin{cases} -(\varepsilon ^{2}a+\varepsilon b{\int _{\mathbb{R}^{3}}}{ \vert \nabla v \vert } ^{2})\Delta v+V(x)v=K(x) \vert v \vert ^{p-1}v \quad \mbox{in }\mathbb{R}^{3}, \\ v>0, \quad v\in H^{1}(\mathbb{R}^{3}), \end{cases} $$ where ε>0 $\varepsilon >0$ is a small parameter, a,b>0 $a,b>0$ are constants, 3