A critical exponent for Hénon type equation on the hyperbolic space

In this paper, we discuss a critical exponent with respect to the stability of solutions to an Hénon type equation on the hyperbolic space. In Euclidean space, there exists a critical exponent on stable solutions of an Hénon type equation. In particular, this exponent is called the Joseph–Lundgren exponent in case of the Lane–Emden equation. However, on the hyperbolic space, the Lane–Emden equation admits no such critical exponent on stability. We devote this paper to showing the existence of a critical exponent of Joseph–Lundgren type for a weighted Lane–Emden equation. More precisely, we prove non-existence of non-trivial stable solutions of the Hénon type equation for the subcritical case. Moreover, we show that the Hénon type equation has stable, positive, and radial solutions for the supercritical case.