Sign-Changing Solutions for Planer Kirchhoff Type Problem With Critical Exponential Growth

In the present paper, we study the existence of ground state sign-changing solutions for a class of Kirchhoff type problem \(\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\Omega }|\nabla u|^{2}dx\right) \Delta u=f(u), &{} \quad x\in \Omega ; \\ u=0,&{} \quad x\in \partial \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\left(a,+,b,{\int }_{\mathrm{\Omega }},{|\mathrm{\nabla }u|}^2,d,x\right)\mathrm{\Delta }u=f(u),\hfill & x\in \mathrm{\Omega };\hfill \\ u=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(a, b>0\) $a,b>0$ , \(f\in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})\) $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ satisfies subcritical exponential growth or critical exponential growth in a bounded domain \(\Omega \in {\mathbb {R}}^{2}\) $\mathrm{\Omega }\in {\mathbb{R}}^2$ with a smooth boundary \(\partial \Omega \) $\partial \mathrm{\Omega }$ . In combination with Trudinger-Moser inequality, we prove the existence of least energy sign-changing solution by variational method and obtain its concentration behaviors as \(b\searrow 0\) $b\searrow 0$ in both subcritical case and critical case.