Elliptic Problems Involving Mixed Local-Nonlocal Operator in the Hyperbolic Space

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