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Existence and multiplicity of non-radial sign-changing solutions for a semilinear elliptic equation in hyperbolic space

  • Atanu Manna ORCID logo and Bhakti Bhusan Manna ORCID logo EMAIL logo
Published/Copyright: January 13, 2025

Abstract

In this article, we consider the following problem:

- Δ 𝔹 N u - λ u = | u | p - 1 u , u H 1 ( 𝔹 N ) ,

where 𝔹 N represents the Poincaré ball model of the hyperbolic space, 1 < p < 2 * - 1 = N + 2 N - 2 , λ < ( N - 1 ) 2 4 , N 4 , and H 1 ( 𝔹 N ) denotes the Sobolev space on 𝔹 N . In [D. Ganguly and S. Kunnath, Sign changing solutions of the Brezis–Nirenberg problem in the hyperbolic space, Calc. Var. Partial Differential Equations 50 2014, 1–2, 69–91], Ganguly and Sandeep establish the existence of radial sign-changing solutions for the critical case. In this work, we will discuss the existence and multiplicity of non-radial sign-changing solutions for the sub-critical problem. To establish the presence of sign-changing solutions that are not radial, we will demonstrate the existence of an isometric group action on 𝔹 N and operate inside a permissible space of functions that change signs. We shall also discuss the multiplicity of sign-changing solutions.


Communicated by Maria del Mar Gonzalez


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Received: 2024-07-28
Revised: 2024-11-05
Published Online: 2025-01-13
Published in Print: 2025-06-01

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