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Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 155 / Issue 5 / October 2025
- Published online by Cambridge University Press:
- 27 February 2024, pp. 1708-1746
- Print publication:
- October 2025
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- Article
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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.
