Bubbling nodal solutions for a large perturbation of the Moser-Trudinger equation on planar domains

Abstract

In this work we study the existence of nodal solutions for the problem $$-\Delta u=\lambda u{e}^{u^2+|u{|}^p}\text{ in }\Omega ,u=0\text{ on }\partial \Omega ,$$ where $\Omega \subseteq {\mathbb{R}}^2$ is a bounded smooth domain and $p\to 1^{+}$. If $\Omega$ is ball, it is known that the case $p=1$ defines a critical threshold between the existence and the non-existence of radially symmetric sign-changing solutions. In this work we construct a blowing-up family of nodal solutions to such problem as $p\to 1^{+}$, when $\Omega$ is an arbitrary domain and $\lambda$ is small enough. As far as we know, this is the first construction of sign-changing solutions for a Moser-Trudinger critical equation on a non-symmetric domain.