  This paper deals with the global compactness and multiplicity of positive
solutions to problems of the type $$ -\Delta_{\mathbb B^N} u -\lambda u=a(x)
|u|^{2^*-2}u+f(x) \quad\text{in } \mathbb B^N, \quad u\in H^1(\mathbb B^N),$$
  where $\mathbb B^N$ denotes the ball model of the hyperbolic space of
dimension $N\geq 4$, $2^*=\frac{2N}{N-2}$,
$\frac{N(N-2)}{4}<\lambda<\frac{(N-1)^2}{4}$ and $f\in H^{-1}(\mathbb B^N)$
($f\not\equiv 0$) is a non-negative functional in the dual space of
$H^1(\mathbb B^N)$. The potential $a\in L^\infty(\mathbb B^N)$ is assumed to be
strictly positive, such that $\lim_{ d(x,0)\to \infty}a(x)=1$, where $d(x,0)$
denotes the geodesic distance. We establish profile decomposition of the
associated functional. We show that concentration takes place along two
different profiles, namely along hyperbolic bubbles and localized Aubin-Talenti
bubbles. For $f=0$ and $a\equiv 1$, profile decomposition was studied by Bhakta
and Sandeep [Calc. Var. PDE, 2012]. However, due to the presence of $a(.)$, an
extension of profile decomposition to the present set-up is highly nontrivial
and requires several delicate estimates and geometric arguments concerning the
isometry group (M\"obius group) of the hyperbolic space. Further, using the
decomposition result, we derive various energy estimates involving the
interacting hyperbolic bubbles and hyperbolic bubbles with localized
Aubin-Talenti bubbles. Finally, combining these estimates with topological and
variational arguments, we establish a multiplicity of positive solutions in the
cases: $a\geq 1$ and $a<1$ separately. The equation studied in this article can
be thought of as a variant of a scalar-field equation with a critical exponent
in the hyperbolic space, although such a critical exponent problem in the
Euclidean space $\mathbb{R}^N$ has only a trivial solution when $f \equiv 0,$
$a(x)\equiv1$ and $\lambda < 0.$
