Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation - Δ u + u = u 2 ∗ - 1 + λ u q - 1 in R N , ( P λ ) where N ≥ 3 is an integer, 2 ∗ = 2 N N - 2 is the Sobolev critical exponent, 2 < q < 2 ∗ and λ > 0 is a parameter. It is known that as λ → 0 , after a rescaling the ground state solutions of ( P λ ) converge to a particular solution of the critical Emden-Fowler equation - Δ u = u 2 ∗ - 1 . We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension N = 3 , N = 4 or N ≥ 5 . We also discuss a connection of these results with a mass constrained problem associated to ( P λ ) . Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the L 2 norm of the groundstates.