2024/09/16 by Diksha Gupta, Gupta, Diksha, K. Sreenadh +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Functional Analysis (math.FA) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2409.10236
openalex publication_date 2024/09/16 · openalex created_date 2024/10/25 · openalex updated_date 2026/07/28
In this paper, we explore the positive solutions of the following nonlinear Choquard equation involving the green kernel of the fractional operator (-Δ_\mathbbBN)-α/2 in the hyperbolic space \beginaligned -Δ_\mathbbBN u - λu amp;= [(- Δ_\mathbbBN)^-\fracα2|u|p]|u|p-2u, \endaligned where Δ_\mathbbBN denotes the Laplace-Beltrami operator on \mathbbBN, λ≤ ((N-1)2)/(4), 1 < p < 2^*α = (N+α)/(N-2), 0 < α< N, N ≥ 3, 2^*α is the critical exponent in the context of the Hardy-Littlewood-Sobolev inequality. This study is analogous to the Choquard equation in the Euclidean space, which involves the non-local Riesz potential operator. We consider the functional setting within the Sobolev space H1(\mathbbBN), employing advanced harmonic analysis techniques, particularly the Helgason Fourier transform and semigroup approach to fractional Laplacian. Moreover, the Hardy-Littlewood-Sobolev inequality on complete Riemannian manifolds, as developed by Varopoulos, is pivotal in our analysis. We prove an existence result for the above problem in the subcritical case. Moreover, we also demonstrate that solutions exhibit radial symmetry, and establish the regularity properties.