$n$Kirchhoff type equations with exponential nonlinearities

In this article, we study the existence of non-negative solutions of the class of non-local problem of $n$-Kirchhoff type $$\{\begin{array}{l}-m({\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^n){\mathrm{\Delta }}_{n}u=f(x,u)\text{in}\mathrm{\Omega },u=0\text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }\subset {\mathbf{R}}^n$ is a bounded domain with smooth boundary, $n\ge 2$ and $f$ behaves like $e^{\mathrm{\mid }u{\mathrm{\mid }}^{\frac{n}{n-1}}}$ as $\mathrm{\mid }u\mathrm{\mid }\to \mathrm{\infty }$. Moreover, by minimization on the suitable subset of the Nehari manifold, we study the existence and multiplicity of solutions, when $f(x,t)$ is concave near $t=0$ and convex as $t\to \mathrm{\infty }\mathrm{.}$