It is establish existence and multiplicity of solutions for nonlocal elliptic problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following Choquard type problem: { − Δ u + V ( x ) u = μ ( I α 1 ∗ | u | q ) | u | q − 2 u − λ ( I α 2 ∗ | u | p ) | u | p − 2 u in R N , u ∈ H 1 ( R N ) , where p > q , λ , μ > 0 , α 1 ≤ α 2 ; α 1 , α 2 ∈ ( 0 , N ) , N ≥ 3 ; p ∈ ( 2 α 2 , 2 α 2 ∗ ) ; q ∈ ( 2 α 1 , 2 α 1 ∗ ) , 2 α j = ( N + α j ) / N and 2 α j ∗ = ( N + α j ) / ( N − 2 ) , j = 1 , 2 . Here we employ some variational arguments together with the Nehari method and the nonlinear Rayleigh quotient. The main feature in the present work is to find a sharp μ n > 0 and λ ∗ , λ ∗ > 0 such that our main problem admits at least two solutions for each μ > μ n where λ ∈ ( 0 , min ( λ ∗ , λ ∗ ) ) . The main difficulty here is to prove that the infimum associated to the energy functional restricted to the Nehari set is a weak solution for our main problem. This phenomenon occurs since the fibering maps for the associated energy functional have inflection points. Furthermore, we prove a nonexistence result for our main problem for each μ < μ n and λ > 0 .