For each s in (1/2,1), every finite-mass solution of (-Δ)^s u = e^u on R equals one explicit profile Q_s up to translation and scaling.
Beckner , Functionals for multilinear fractional embedding , Acta Math
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Existence and Uniqueness for the Fractional Gelfand Equation in $\mathbb{R}$
For each s in (1/2,1), every finite-mass solution of (-Δ)^s u = e^u on R equals one explicit profile Q_s up to translation and scaling.