This paper aims to study the eigenvalue problems of a mixed local and nonlocal operator in the exterior of a nonempty, bounded, simply connected domain \(\varOmega \subset {\mathbb {R}}^N\) with Lipschitz boundary \(\partial \varOmega \ne \emptyset \) . By employing the variational methods combined with the Ljusternik-Schnirelmann principle, we prove the existence of a non-decreasing sequence of eigenvalues. In particular, we prove the principal eigenvalue is simple and isolated. We establish the positivity of the first eigenfunction by obtaining a strong maximum principle. The results obtained here are new even for the case \(p=2\) .
