New solutions for Trudinger–Moser critical equations in R2

Abstract

Let Ω be a bounded, smooth domain in R2. We consider critical points of the Trudinger–Moser type functional Jλ(u) = 12 Ω |∇u|2 − λ2 Ω eu2 in H1 0 (Ω), namely solutions of the boundary value problem u + λueu2 = 0 with homogeneous Dirichlet boundary conditions, where λ > 0 is a small parameter. Given k 1 we find conditions under which there exists a solution uλ which blows up at exactly k points in Ω as λ→0 and Jλ(uλ)→2kπ. We find that at least one such solution always exists if k = 2 and Ω is not simply connected. If Ω has d 1 holes, in addition d +1 bubbling solutions with k = 1 exist. These results are existence counterparts of one 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, including this one as a prototype case.