Existence and Uniqueness for the Fractional Gelfand Equation in $\mathbb{R}$

We prove existence, symmetry and uniqueness of solutions to the fractional Gelfand equation (-Δ)^s u = e^u \quad in $\mathbb{R$} \quad with \quad \int_{\mathbb{R}} e^u dx < +\infty for all exponents $s\in (\frac{1}{2},1)$. Furthermore, we show $u$ has finite Morse index and that its linearized operator is nondegenerate. Our arguments are based on a fixed point scheme in terms of the function $v=\sqrt{e^u}$ and we devise a nonlocal shooting method involving (locally) compact nonlinear maps. We also study existence, symmetry and uniqueness of solutions to $(-\mathrm{\Delta }{)}^{s}u=K{e}^u$ in $\mathbb{R}$ with $K{e}^u\in L^1(\mathbb{R})$ for a general class of positive, even and monotone-decreasing functions $K>0$.