LAUSR

In this work we study the existence of nodal solutions for the problem $$\begin{aligned} -\Delta u = \lambda u e^{u^2+|u|^p} \quad \text { in }\quad \Omega ,\quad u = 0 \quad \text { on }\partial \Omega , \end{aligned}$$ - Δ u = λ u e u 2 + | u | p in Ω , u = 0 on ∂ Ω , where $$\Omega \subseteq \mathbb {R}^2$$ Ω ⊆ R 2 is a bounded smooth domain and $$p\rightarrow 1^+$$ p → 1 + . If $$\Omega $$ Ω is a ball, it is known that the case $$p=1$$ 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 $$p\rightarrow 1^+$$ p → 1 + , when $$\Omega $$ Ω is an arbitrary domain and $$\lambda $$ 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.