Finite Morse Index Solutions of the Fractional Henon–Lane–Emden Equation with Hardy Potential

Abstract

In this paper, we study the fractional Henon–Lane–Emden equation associated with Hardy potential \[ (-\Delta)^{s} u - \gamma |x|^{-2s} u = |x|^a |u|^{p-1} u \quad \textrm{in $\mathbb{R}^{n}$}. \] Extending the celebrated result of [14], we obtain a classification result on finite Morse index solutions to the fractional elliptic equation above with Hardy potential. In particular, a critical exponent $p$ of Joseph–Lundgren type is derived in the supercritical case studying a Liouville type result for the $s$-harmonic extension problem.