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This paper deals with the global compactness and multiplicity of positivesolutions 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 ofdimension $N\geq 4$, $2^*=\frac{2N}{N-2}$,$\frac{N(N-2)}{4}<\lambda<\frac{(N-1)^2}{4}$ 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 bestrictly 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 theassociated functional. We show that concentration takes place along twodifferent profiles, namely along hyperbolic bubbles and localized Aubin-Talentibubbles. For $f=0$ and $a\equiv 1$, profile decomposition was studied by Bhaktaand Sandeep [Calc. Var. PDE, 2012]. However, due to the presence of $a(.)$, anextension of profile decomposition to the present set-up is highly nontrivialand requires several delicate estimates and geometric arguments concerning theisometry group (M\"obius group) of the hyperbolic space. Further, using thedecomposition result, we derive various energy estimates involving theinteracting hyperbolic bubbles and hyperbolic bubbles with localizedAubin-Talenti bubbles. Finally, combining these estimates with topological andvariational arguments, we establish a multiplicity of positive solutions in thecases: $a\geq 1$ and $a<1$ separately. The equation studied in this article canbe thought of as a variant of a scalar-field equation with a critical exponentin the hyperbolic space, although such a critical exponent problem in theEuclidean space $\mathbb{R}^N$ has only a trivial solution when $f \equiv 0,$$a(x)\equiv1$ and $\lambda < 0.$

Global compactness result and multiplicity of solutions for a class of critical exponent problem in the hyperbolic space