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This article concerns the existence of positive solutions for the second order equation involving a nonlocal term −∆u = γ(−∆)−1u + |u| p−1u, under Dirichlet boundary conditions. We prove the existence of positive solutions depending on the positive real parameter γ > 0, and up to the critical value of the exponent p, i.e. when 1 < p ≤ 2 ∗ − 1, where 2∗ = 2N N−2 is the critical Sobolev exponent. For p = 2∗ − 1, this leads us to a Brezis-Nirenberg type problem, cf. [5], but, in our particular case, the linear term is a nonlocal term. The effect that this nonlocal term has on the equation changes the dimensions for which the classical technique based on the minimizers of the Sobolev constant, that ensures the existence of positive solution, going from dimensions N ≥ 4 in the classical Brezis-Nirenberg problem, to dimensions N ≥ 7 for this nonlocal problem. This research was partially supported by the Ministry of Economy and Competitiveness of Spain, and by the FEDER under research projects MTM2016-80618-P and PID2019-106122GB-I00. P. Alvarez-Caudevilla was also supported by the Ministry of Economy and Competitiveness of Spain under research project RYC-2014-15284.