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

In this paper, we prove the existence of nontrivial solutions and Nehari-type ground-state solutions for the following Kirchhoff-type elliptic equation: {-m(parallel to del u parallel to(2)(2))Delta u = f(x, u), in Omega, u = 0, on partial derivative Omega, where Omega subset of R-2 is a smooth bounded domain, m : R+ -> 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.