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.
Categories: math.AP