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

In this work we study the existence of nodal solutions for the problem -Δu=λueu2+|u|pinΩ,u=0on∂Ω,where Ω ⊆ R2 is a bounded smooth domain and p→ 1 +. If Ω is a 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→ 1 +, when Ω is an arbitrary domain and λ 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.