Localized concentration of semiclassical solutions for double phase problems with nonlocal reaction

Abstract This paper focuses on the study of multiplicity and localized concentration properties of positive solutions for the following singularly perturbed double phase problem with nonlocal Choquard reaction $$\begin{aligned} \left\{ \begin{array}{ll} -\epsilon ^{p}\Delta _{p} u-\epsilon ^{q}\Delta _{q} u +V(x)(|u|^{p-2}u+|u|^{q-2}u)\\ \quad =\epsilon ^{\mu -N}\left( \frac{1}{|x|^{\mu }}*G(u)\right) g(u),& \hbox {in}~\mathbb {R}^{N},\\ u\in W^{1,p}(\mathbb {R}^{N})\cap W^{1,q}(\mathbb {R}^{N}),u>0, & \hbox {in}~\mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$ - ϵ p Δ p u - ϵ q Δ q u + V ( x ) ( | u | p - 2 u + | u | q - 2 u ) = ϵ μ - N 1 | x | μ ∗ G ( u ) g ( u ) , in R N , u ∈ W 1 , p ( R