Multiple normalized solutions for a class of dipolar Gross-Pitaveskii equation with a mass subcritical perturbation

In this paper, we study the existence of multiple normalized solutions to the following dipolar Gross-Pitaveskii equation with a mass subcritical perturbation {align*} \{ {array}{lll} -1/2Δu+μu+V(ε x)u + λ_1 |u|²u + λ_2(K|u|²)u + λ_3|u|ᵖ⁻²u = 0, \;&in\; R³,\\ ∫R^3 |u|²dx = a², {array}. {align*} where a,ε>0, 2<p<10/3, μ∈ R denotes the Lagrange multiplier, λ₃<0, (λ₁,λ₂) ∈ (λ₁,λ₂) ∈ R²:λ₁<4π/3λ₂≤ 0\; or\; λ₁<-8π/3λ₂≤ 0, $V(x)$ is an external potential, stands for the convolution, K(x)=1-3cos²θ(x)|x|³ and $θ(x)$ is the angle between the dipole axis determined by $(0,0,1)$ and the vector x. Under some assumptions of V, we use variational methods to prove that the number of normalized solutions is not less than the number of global minimum points of V if ε> 0 is sufficiently small.