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

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