This article deals with the study of the following critical exponent problem (Formula presented.) where Hn is the n-dimensional hyperbolic space, n ≥ 4, p = 2n/(n − 2) is the critical exponent, λ,μ > 0 are real parameters, ΔHn denotes the Laplace–Beltrami operator on Hn and g(x) is a real valued potential function on Hn. Using variational methods, we establish the existence of positive ground state solution to (Pλ,μ) and study the convergence of these solutions when μ approaches +∞. © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group.