Asymptotic Behavior Related to Cheeger Constant for Solutions of an Exponentially Growth Equation

We show the existence of a positive solution \(u_{p}\) $u_p$ for a Dirichlet p-Laplacian problem with nonlinearity involving an exponential term that can be supercritical. We determine the asymptotic behavior of \(u_{p}\) $u_p$ as \(p\rightarrow 1^{+}\) $p\to 1^{+}$ and \(p\rightarrow +\infty \) $p\to +\infty$ , which are related to the Cheeger constant and the distance function to the boundary, respectively. Furthermore, we also prove a nonexistence result concerning the range of the parameter \(\lambda \) $\lambda$ .