We study the existence of solution to the problemwhere Q ≥ 0, κ ∈ (0, ∞) and n ≥ 3. Using ODE techniques Martinazzi for n = 6 and Huang-Ye for n = 4m + 2 proved the existence of solution to the above problem with Q ≡ const > 0 and for every κ ∈ (0, ∞). We extend these results in every dimension n ≥ 5, thus completely answering the problem opened by Martinazzi. Our approach also extends to the case in which Q is non-constant, and under some decay assumptions on Q we can also treat the cases n = 3 and 4.

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We study the existence of solutions to the problemwhere Q = (n − 1)! or Q = −(n − 1)!. Extending the works of Wei-Ye and Hyder-Martinazzi to arbitrary odd dimension n ≥ 3 we show that to a certain extent the asymptotic behavior of u and the constant V can be prescribed simultaneously. Furthermore if Q = −(n − 1)! then V can be chosen to be any positive number. This is in contrast to the case n = 3, Q = 2, where Jin-Maalaoui-Martinazzi-Xiong showed that necessarily V ≤ |S 3 |, and to the case n = 4, Q = 6, where C-S. Lin showed that V ≤ |S 4 |.

In this article we study the nonlocal equationwhich arises in the conformal geometry. Inspired by the previous work of C. S. Lin and L. Martinazzi in even dimension and T. Jin, A. Maalaoui, L. Martinazzi, J. Xiong in dimension three we classify all solutions to the above equation in terms of their behavior at infinity.

We study finite Morse index solutions to the non-local Gelfand–Liouville problem $$\begin{align*}& (-\Delta)^su=e^u\quad\textrm{in}\quad{{\mathbb{R}}^n}, \end{align*}$$for every $s\in (0,1)$ and $n>2s$. Precisely, we prove non-existence of finite Morse index solutions whenever the singular solution $$\begin{align*} &u_{n,s}(x)=-2s\log|x|+\log \left(2^{2s}\frac{\Gamma(\frac{n}{2})\Gamma(1+s)}{\Gamma(\frac{n-2s}{2})}\right)\end{align*}$$is unstable.