  We study the a priori estimates,existence/nonexistence of radial sign
changing solution, and the Palais-Smale characterisation of the problem
$-\De_{\Bn}u - \la u = |u|^{p-1}u, u\in H^1(\Bn)$ in the hyperbolic space $\Bn$
where $1<p\leq\frac{N+2}{N-2}$. We will also prove the existence of sign
changing solution to the Hardy-Sobolev-Mazya equation and the critical Grushin
problem.
