Author:
Ganguly Debdip,Gupta Diksha,Sreenadh K.
Abstract
The paper is concerned with positive solutions to problems of the type
\[ -\Delta_{\mathbb{B}^{N}} u - \lambda u = a(x) |u|^{p-1}\;u + f \text{ in }\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, \]
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.
Publisher
Cambridge University Press (CUP)