The existence and local uniqueness of the eigenfunctions of the non-linear operator ${\mathrm{\Delta }}_H{u}^n$ in the hyperbolic Poincaré half-plane

In this article we find locally an eigenfunctions for a particular nonlinear hyperbolic differential operator ${\mathrm{\Delta }}_H{u}^n$, where ${\mathrm{\Delta }}_H$ is the hyperbolic Laplacian in the half-plane of Poincairé. 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 Poincairé half-plane, which could describe a possible one-dimensional physical model.