Analog of the blow-up solutions to a discrete second-order equation of the Emden–Fowler type

We consider a discrete analog of the differential equation of the Emden–Fowler type
$$ \Delta^2v(k)=-k^s (\Delta v(k))^3, $$
where $k \to \infty$, $s \ne 1$, $s\in \mathbb{R}$, $\Delta v(k)=v(k+1)-v(k)$. It is a discrete analog of the second-order nonlinear equation $y''(x)=y^s(x)$. We prove the existence of an approximate solution of the form $V(k)=\pm\dfrac{\sqrt{2s+2}}{s-1} k^{(1-s)/2}$ and a nontrivial solution tending to $0$ as $k \to \infty$.