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

Given a sufficiently symmetric domain $\mathrm{\Omega }⋐{\mathbb{R}}^2$, for any $k\in \mathbb{N}\setminus \{0\}$ and $\beta >4\pi k$ we construct blowing-up solutions $(u_{\epsilon })\subset H_0^1(\mathrm{\Omega })$ to the Moser-Trudinger equation such that as $\epsilon \downarrow 0$, we have $|\nabla u_\varepsilon|{L^2}^2\to \beta$,$u\varepsilon \rightharpoonup u_0$in$H^1_0$where$u_0$ is a sign-changing solution of the Moser-Trudinger equation and $u_{\epsilon }$ develops $k$ positive spherical bubbles, all concentrating at $0\in \mathrm{\Omega }$. These $3$ features (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 (J. Eur. Math. Soc. (JEMS) 22 (2020)).