Nodal solution for critical Kirchhoff-type equation with fast increasing weight in $\mathbb{R}^{2}$

Abstract In this paper, we investigate the existence of a least-energy sign-changing solutions for following Kirchhoff-type equation: $$ - \biggl(1+b \int _{\mathbb{R}^{2}} K(x) \vert \nabla u ^{2}\,dx \biggr) \operatorname{div} \bigl(K(x)\nabla \bigr)=K(x)f(u),\quad x\in \mathbb{R}^{2}, − ( 1 + b ∫ R 2 K x ) | ∇ u d div = f , ∈ where f has exponential subcritical or critical growth in sense Trudinger–Moser inequality. By using constrained variational methods, combining deformation lemma and Miranda’s theorem, prove solution. Moreover, also that solution exactly two nodal domains.