  The paper is concerned with positive solutions to problems of the type
  \begin{equation*}
  -\Delta_{\mathbb{B}^N} u - \lambda u = a(x) |u|^{p-1}\;u \, + \, f \,
\;\;\text{in}\;\mathbb{B}^{N}, \quad
  u \in H^{1}{(\mathbb{B}^{N})},
  \end{equation*}
  where $\mathbb{B}^N$ denotes the hyperbolic space,
$1<p<2^*-1:=\frac{N+2}{N-2}$, $\;\lambda < \frac{(N-1)^2}{4}$, and $f \in
H^{-1}(\mathbb{B}^N)$ ($f \not\equiv 0$) is a non-negative functional. The
potential $a\in L^\infty(\mathbb{B}^N)$ is assumed to be strictly positive,
such that $\lim_{d(x, 0) \rightarrow \infty} a(x) \rightarrow 1,$ where $d(x,
0)$ denotes the geodesic distance. First, the existence of three positive
solutions is proved under the assumption that $a(x) \leq 1$. Then the case
$a(x) \geq 1$ is considered, and the existence of two positive solutions is
proved. In both cases, it is assumed that $\mu( \{ x : a(x) \neq 1\}) > 0.$
Subsequently, we establish the existence of two positive solutions for $a(x)
\equiv 1$ and prove asymptotic estimates for positive solutions using
barrier-type arguments. The proofs for existence combine variational arguments,
key energy estimates involving hyperbolic bubbles.
