Blow-up analysis for nodal radial solutions in Moser-Trudinger critical equations in $R^2$

In this paper we consider sign-changing radial solutions u " to the ( 2 1+" 1u = ueu +|u| u=0 in B on @ B, and we study their asymptotic behaviour as " & 0. We show that when u " = u " (r) has k interior zeros, it exhibits a multiple blow-up behaviour in the first k nodal sets while it converges to the least energy solution of the problem with " = 0 in the (k + 1)-th one. We also prove that in each concentration set, with an appropriate scaling, u " converges to the solution of the classical Liouville problem in R2 . −∆u = λue^{u^2} +|u|^{1+ε} in B, u =0 on ∂B, and we study their asymptotic behaviour as ε & 0. We show that when u ε = u ε (r) has k interior zeros, it exhibits a multiple blow–up behaviour in the first k nodal sets while it converges to the least energy solution of the problem with ε = 0 in the (k + 1)–th one. We also prove that in each concentration set, with an appropriate scaling, u ε converges to the solution of the classical Liouville problem in R 2 .