Existence of Positive Solutions for Local-Nonlocal Elliptic Problems in Exterior Domains

Abstract

In this paper, we investigate the existence of positive solutions for a class of elliptic problems driven by a mixed local-nonlocal operator in exterior domains. Specifically, we consider the problem governed by the Laplacian and the fractional Laplacian, focusing on the critical Sobolev exponent range and the interplay between local and nonlocal terms. Our approach, inspired by variational methods and the foundational work of Benci and Cerami (see [6]), extends classical existence results to a setting where the exterior geometry and fractional effects play a crucial role.