期刊
JOURNAL OF FUNCTIONAL ANALYSIS
卷 258, 期 2, 页码 421-457出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jfa.2009.06.018
关键词
Trudinger-Moser inequality; Blowing-up solutions; Singular perturbations
类别
Let Omega be a bounded, smooth domain in R-2. We consider critical points of the Trudinger-Moser type functional J(lambda) (u) = 1/2 integral(Omega)vertical bar del u vertical bar(2) - lambda/2 integral(Omega)e(u2) in H-0(1)(Omega), namely solutions of the boundary value problem Delta u + lambda ue(u2) = 0 with homogeneous Dirichlet boundary conditions, where lambda > 0 is a small parameter. Given k >= 1 we find conditions under which there exists a solution u(lambda) which blows up at exactly k points in Omega as lambda -> 0 and J(lambda)(u(lambda)) -> 2k pi. We find that at least one such solution always exists if k = 2 and Omega is not simply connected. If Omega 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. (C) 2009 Elsevier Inc. All rights reserved.
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