Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth

Abstract

In this paper, we prove the existence of nontrivial solutions and Nehari-type ground-state solutions for the following Kirchhoff-type elliptic equation: $$\begin{aligned} {\left\{ \begin{array}{ll} -m(\Vert \nabla u\Vert _2^2)\Delta u=f(x,u), \;\;&{} \text{ in } \ \ \Omega ,\\ u= 0, \;\;&{} \text{ on } \ \ \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset \mathbb {R}^2$$ is a smooth bounded domain, $$m:\mathbb {R}^+\rightarrow \mathbb {R}^+$$ is a Kirchhoff function, and f has critical exponential growth in the sense of Trudinger–Moser inequality. We develop some new approaches to estimate precisely the minimax level of the energy functional and prove the existence of Nehari-type ground-state solutions and nontrivial solutions for the above problem. Our results improve and extend the previous results. In particular, we give a more precise estimation than the ones in the existing literature about the minimax level, and also give a simple proof of a known inequality due to P.L. Lions.