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 \[ \begin{cases} -Δu=λue^{u^2+|u|^{1+ε}}&\text{ in }B,\\ u=0&\text{ on }\partial B. \end{cases} \] and we study their asymptotic behaviour as $\epsilon \searrow 0$, $\epsilon >0$. 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 $R^2$.