Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space

In this study, we explore the positive solutions of a nonlinear Choquard equation involving the Green kernel of the fractional operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">B</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>α</m:mi> <m:mo>⁄</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> {(-{Δ }_{{{B}}N})}-α /2 in the hyperbolic space, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">B</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> </m:msub> </m:math> {Δ }_{{{B}}N} represents the Laplace-Beltrami operator on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">B</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> {{B}}N , with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>N</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> α ∈ (0,N) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> N≥ 3 . This study is analogous to the Choquard equation in the Euclidean space, which involves the nonlocal Riesz potential operator. We consider the functional setting within the Sobolev space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">B</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {H}¹({{B}}N) , employing advanced harmonic analysis techniques, particularly the Helgason Fourier transform and semigroup approach to the 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 problem ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>F</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> </m:mrow> </m:msub> </m:math> {F}α ,λ ) in the subcritical case. Moreover, we also demonstrate that solutions exhibit radial symmetry, and establish the regularity properties.