  In this paper, the following problem in the hyperbolic space $\mathbb{B}^N$
will be considered \begin{equation*} -\Delta_{\mathbb{B}^N} u=f(x,u),
\mathrm{in} \ \mathbb{B}^N.\eqno{(1)} \end{equation*} where,
$\Delta_{\mathbb{B}^N}$ denotes the Laplace Beltrami operator on
$\mathbb{B}^N$. And this problem can be converted into the following Euclidean
problem \begin{equation*} \begin{cases} -\operatorname{div}(K(x) \nabla u)=4
K(x)^{\frac{N}{N-2}}f(x,u), &\mathrm{in} \ \mathbb{B}^N, \\ u(0)=0,
&\mathrm{on}\ \partial\mathbb{B}^N, \end{cases}\eqno{(2)} \end{equation*}
where, $K(x):=1/\left(1-|x|^2\right)^{N-2}.$ Then, the existence of solution of
problem (1) can be obtained by studying the existence of solution of problem
(2). We will equip problem (2) with a weighted Sobolev space and prove the
compact embedding theorem and the concentration compactness principle for the
weighted Sobolev space. And we will prove that the maximum principle holds for
the operator $-\operatorname{div}(K(x) \nabla u)$.
  When $f(x,u)=|u|^{2^*-2} u+\lambda u^{q-2}u$, $\lambda>0$, $1<q<2^{\ast}$,
using the variational method, the compact embedding theorem, the concentration
compactness principle and the maximum principle, the existence of nonradial
solutions of problem (2) will be obtained.
