Normalized solutions for a Schrödinger equation with critical growth in RN

In this paper we study the existence of normalized solutions to the following nonlinear Schrödinger equation with critical growth - Δ u = λ u + f ( u ) , in R N , u > 0 , ∫ R N | u | 2 d x = a 2 , where a > 0 , λ ∈ R and f has an exponential critical growth when N = 2 , and f ( t ) = μ | t | q - 2 t + | t | 2 ∗ - 2 t with q ∈ ( 2 + 4 N , 2 ∗ ) , μ > 0 and 2 ∗ = 2 N N - 2 when N ≥ 3 . Our main results complement some recent results for N ≥ 3 and it is totally new for N = 2 .