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

TL;DR

This paper investigates the existence of solutions for nonlinear elliptic equations with mixed local and nonlocal operators in hyperbolic space, using variational methods to handle subcritical and critical cases.

Contribution

It introduces a novel analysis of elliptic equations involving a combined local-nonlocal operator in hyperbolic space, extending existing theories to this setting.

Findings

Existence of weak solutions for subcritical nonlinearities

Existence of weak solutions for critical nonlinearities

Application of variational methods in hyperbolic geometry

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 $\mathbb{B}^N$. By employing variational methods, we address both subcritical and critical nonlinearities, establishing the existence of weak solutions under appropriate conditions.