Critical points of the Trudinger–Moser trace functional with high energy levels

Let Ω   be a bounded domain in R2R2 with smooth boundary. In this paper we are concerned with the existence of critical points for the supercritical Trudinger–Moser trace functionalequation(0.1)∫∂Ωekπ(1+μ)u2 in the set {u∈H1(Ω):∫Ω(|∇u|2+u2)dx=1}, where k⩾1k⩾1 is an integer and μ>0μ>0 is a small parameter. For any integer k⩾1k⩾1 and for any μ>0μ>0 sufficiently small, we prove the existence of a pair of k-peaks constrained critical points of the above problem.