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

Given a sufficiently symmetric domain Ω⋐R2, for any k∈N∖{0} and β>4πk we construct blowing-up solutions (uε)⊂H01(Ω) to the Moser–Trudinger equation such that as ε↓0, we have ‖∇uε‖Ljavax.xml.bind.JAXBElement@57b4bd332→β, uε⇀u0 in H01 where u0 is a sign-changing solution of the Moser–Trudinger equation and uε develops k positive spherical bubbles, all concentrating at 0∈Ω. These 3 features (lack of quantization, non-zero weak limit and bubble clustering) stand in sharp contrast to the positive case (uε>0) studied by the second author and Druet [8].