2021/04/11 by Martinazzi, Luca, Thizy, Pierre-Damien, Vétois, Jérôme
#35J60 #35J91 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2104.04959
Given a sufficiently symmetric domain Ω\Subsetℝ2, for any k∈ ℕ∖ \0\ and β>4πk we construct blowing-up solutions (uε)⊂ H10(Ω) to the Moser-Trudinger equation such that as ε\downarrow 0, we have ‖∇ uε‖L22→ β, uε \rightharpoonup u0 in H10 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 (J. Eur. Math. Soc. (JEMS) 22 (2020)).