Existence and non-existence of nonradial solutions to an elliptic equation on hyperbolic space with critical…

Abstract

In this article, we study the semi-linear elliptic problem: −Δ_𝔹^N u − λu = |u|^p−1u, u ∈ H¹(𝔹^N), where N ≥ 3, λ> (N(N−2))/4, and 1 < p ≤ 2^*−1. Here, 𝔹^N represents the Poincaré ball model of the hyperbolic space, and H¹(𝔹^N) denotes the Sobolev space on 𝔹^N. In the critical case p = 2^*−1, with λ< ((N−1)²)/4 and N ≥ 4, we establish the existence and multiplicity of nonradial sign-changing solutions. Moreover, our result answers an open question: ``the existence of sign-changing solutions in the lower dimensions 4 ≤ N ≤ 6". We also prove a partial non-existence result for a large class of symmetric solutions when λ> ((N−1)²)/4, based on a quantitative orbit packing condition.

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BibTeX

@misc{manna2026existence,
  title = {{Existence and non-existence of nonradial solutions to an elliptic equation on hyperbolic space with critical and subcritical nonlinearities}},
  author = {Atanu Manna and Bhakti Bhusan Manna},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.25683},
  eprint = {2606.25683},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.25683},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Existence and non-existence of nonradial solutions to an elliptic equation on hyperbolic space with critical and subcritical nonlinearities
AU  - Atanu Manna
AU  - Bhakti Bhusan Manna
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2606.25683
UR  - https://arxiv.org/abs/2606.25683
AB  - In this article, we study the semi-linear elliptic problem: −Δ_𝔹^N u − λu = |u|^p−1u, u ∈ H^1(𝔹^N), where N ≥ 3, λ> (N(N−2))/4, and 1 < p ≤ 2^*−1. Here, 𝔹^N represents the Poincaré ball model of the hyperbolic space, and H^1(𝔹^N) denotes the Sobolev space on 𝔹^N. In the critical case p = 2^*−1, with λ< ((N−1)^2)/4 and N ≥ 4, we establish the existence and multiplicity of nonradial sign-changing solutions. Moreover, our result answers an open question: ``the existence of sign-changing solutions in the lower dimensions 4 ≤ N ≤ 6". We also prove a partial non-existence result for a large class of symmetric solutions when λ> ((N−1)^2)/4, based on a quantitative orbit packing condition.
ER  -

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