We classify finite Morse index solutions of the following Gelfand- Liouville equation (-Δ)su = eu in Rn, for 1 < s < 2 and s = 2 via a novel monotonicity formula and technical blowdown analysis. We show that the above equation does not admit any finite Morse index solution u with (-Δ) s2 u vanishes at infinity provided that n > 2s and Γ2( n+2s 4 ) Γ2( n-2s 4 ) < Γ( n 2 )Γ(1 + s) Γ( n-2s 2 ) , where Γ is the classical Gamma function. The cases of s = 1 and s = 2 are settled by Dancer and Farina [12, 11] and Dupaigne et al. [16], respectively, using Moser iteration arguments established by Crandall and Rabinowitz [10]. The case of 0 < s < 1 is established by Hyder-Yang in [31] applying arguments provided in [14].