Abstract: We considered an initial boundary value problem of a class of p-Kirchhoff equation with exponential nonlinearity and its 
steady state form. For the evolution equation, we adopted the potential well method. We characterized the asymptotic behaviors of the general global solutions to the problem and the depth of the potential well could be attained by using the properties of potential well and  estimates. For the steady state problem, we gave the existence of a ground state
solution by using Lagrange multiplier method.

Key words: p-Kirchhoff equation, power exponential nonlinearity, ground state solution, asymptotic behavior