Sign-changing blow-up for the Moser–Trudinger equation

Given a sufficiently symmetric domain Omega epsilon R-2, for any k epsilon N\{0} and beta > 4 pi k we construct blowing-up solutions (u(epsilon)) subset of H-1(0)(Omega) to the Moser-Trudinger equation such that as epsilon down arrow 0, we have parallel to del(u epsilon)parallel to(2)(L2) -> beta, u(epsilon) -> u(0) in H-0(1) where u(0) is a sign-changing solution of the Moser-Trudinger equation and uedevelops kpositive spherical bubbles, all concentrating at 0 epsilon Omega. These 3features (lack of quantization, non-zero weak limit and bubble clustering) stand in sharp contrast to the positive case (u(epsilon) > 0) studied by the second author and Druet[8]. (C) 2021 Elsevier Inc. All rights reserved.