Mixed Local-Nonlocal Equations in the Hyperbolic Space and Sharp Sobolev Constants

Abstract

This paper explores the existence of solutions to a class of nonlinear elliptic equations involving a mixed local-nonlocal operator of the form $-{\Delta }_{{\mathbb{B}}^N}+(-{\Delta }_{{\mathbb{B}}^N}{)}^s$, with $0<s<1$, set in the hyperbolic space. By employing variational methods, we address both subcritical and critical nonlinearities, establishing the existence of weak solutions under appropriate conditions. We also compare the best Sobolev constants for the mixed and local operators on ${\mathbb{B}}^N$ with the Euclidean Sobolev constant.