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

Abstract

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

Keywords

• Critical exponential
• Constraint variational
• Trudinger–Moser inequality
• Nodal solution
• Miranda’s theorem
• Deformation lemma