Lie Symmetries of Fundamental Solutions to the Leutwiler-Weinstein Equation

Abstract

AbstractIn this article, we study Lie symmetries to fundamental solutions to the Leutwiler-Weinstein equation $$ Lu:={\Delta} u+\frac{k}{x^{n}}\frac{\partial u}{\partial x^{n}}+\frac{\ell}{(x^{n})^{2}}u=0 $$ L u : = Δ u + k x n ∂ u ∂ x n + ℓ ( x n ) 2 u = 0 in the upper half-space $\mathbb {R}^{n}_{+}$ ℝ + n . Starting from the infinitesimal generators of the equation Lu = 0, we deduce symmetries of the equation Lu = δ(x − x0), and using its invariant solutions, we construct a fundamental solution. As an application, we study a Green functions of the operator in the hyperbolic unit ball.