New solutions for Trudinger–Moser critical equations in R2

AbstractLet 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∫Ωeu2 in H01(Ω), namely solutions of the boundary value problem Δu+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