New solutions for Trudinger–Moser critical equations in R^2

Let Ω be a bounded, smooth domain in R^2. We consider critical points of the Trudinger–Moser type functionals Given k, we find conditions under which there exists a solution which blows up at exactly k points in Ω as the parameter λ→0. We find that at least one such solution always exists if k = 2 and Ω is not simply connected. These results are existence counterparts of a result by Druet in [O. Druet, Multibump analysis in dimension 2: Quantification of blow-up levels, Duke Math. J. 132 (2) (2006) 217–269] which classifies asymptotic bounded energy levels of blow-up solutions for a class of nonlinearities of critical exponential growth.