The existence and local uniqueness of the eigenfunctions of the non-linear operator $ \Delta_H u^{n}$ in the hyperbolic Poincar\'e half-plane

Abstract

In this article we find locally an eigenfunctions for a particular nonlinear hyperbolic differential operator $\Delta_H u^{n}$, where $\Delta_H$ is the hyperbolic Laplacian in the half-plane of Poincair\'e. We have proved that these eigenfunctions are an analytic and non-exact whose coefficients satisfy a specific algebraic recursive rule. The existence of these eigenfunctions allows us to find non-exact solutions respecting the spatial coordinate of nonlinear diffusive PDEs on the Poincair\'e half-plane, which could describe a possible one-dimensional physical model.