In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem$$\left(P_{}\right)\left\{\begin{array}{c}\hfill {ℒ}_{}u=f\left(u\right)\text{in}{ℝ}^3,\\ \hfill u\>0\text{in}{ℝ}^3,\\ \hfill u\in H^1\left({ℝ}^3\right),\end{array}\right.$$( P ε ) ℒ ε u = f ( u ) in IR 3 , u > 0 in IR 3 , u ∈ H 1 ( IR 3 ) , whereε is a small positive parameter, f : ℝ → ℝ is a continuous function,$${ℒ}_{}$$ℒ ε is a nonlocal operator defined by$${ℒ}_{}u=M\left(\frac{1}{}{\int }_{{ℝ}^3}{\left|\nabla u\right|}^2+\frac{1}{{}^3}{\int }_{{ℝ}^3}V\left(x\right)u^2\right)\left[{-}^2\Delta u+V\left(x\right)u\right],$$ℒ ε u = M 1 ε ∫ IR 3 | ∇ u | 2 + 1 ε 3 ∫ IR 3 V ( x ) u 2 [ − ε 2 Δ u + V ( x ) u ] ,M : IR+ → IR+ and V : IR3 → IR are continuous functions which verify some hypotheses.