  In this paper we prove a Pohozaev-type identity for both the problem
$(-\Delta+m^2)^su=f(u)$ in $\mathbb{R}^N$ and its harmonic extension to
$\mathbb{R}^{N+1}_+$ when $0<s<1$. So, our setting includes the
pseudo-relativistic operator $\sqrt{-\Delta+m^2}$ and the results showed here
are original, to the best of our knowledge. The identity is first obtained in
the extension setting and then "translated" into the original problem. In order
to do that, we develop a specific Fourier transform theory for the fractionary
operator $(-\Delta+m^2)^s$, which lead us to define a weak solution $u$ of the
original problem if the identity
\begin{equation}\label{defsola}\int_{\mathbb{R}^N}(-\Delta+m^2)^{s/2}u(-\Delta+m^2)^{s/2}v\dd
x=\int_{ \mathbb{R}^N}f(u)v\dd x\tag{S}\end{equation} is satisfied by all $v\in
H^{s}(\mathbb{R}^N)$. The obtained Pohozaev-type identity is then applied to
prove both a result of nonexistence of solution to the case $f(u)=|u|^{p-2}u$
if $p\geq 2^{*}_s$ and a result of existence of a ground state, if $f$ is
modeled by $\kappa u^3/(1+u^2)$, for a constant $\kappa$. In this last case, we
apply the Nehari-Pohozaev manifold introduced by D. Ruiz. Finally, we prove
that positive solutions of $(-\Delta+m^2)^su=f(u)$ are radially symmetric and
decreasing with respect to the origin, if $f$ is modeled by functions like
$t^\alpha$, $\alpha\in(1,2^{*}_s-1)$ or $t\ln t$.
