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

Abstract: In this paper we consider nodal radial solutions $u_\epsilon$ to the problem [ \begin{cases} -\Delta u=\lambda ue^{u^2+|u|^{1+\epsilon}}&\text{ in }B,\ u=0&\text{ on }\partial B. \end{cases} ] and we study their asymptotic behaviour as $\epsilon\searrow0$, $\epsilon>0$. We show that when $u_\epsilon$ 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 $\epsilon=0$ in the $(k+1)$-th one. We also prove that in each concentration set, with an appropriate scaling, $u_\epsilon$ converges to the solution of the classical Liouville problem in $R^2$.