  In this paper, we consider an elliptic operator obtained as the superposition
of a classical second-order differential operator and a nonlocal operator of
fractional type. Though the methods that we develop are quite general, for
concreteness we focus on the case in which the operator takes the form
$-\Delta+(-\Delta)^s$, with $s\in(0,1)$. We focus here on symmetry properties
of the solutions and we prove a radial symmetry result, based on the moving
plane method, and a one-dimensional symmetry result, related to a classical
conjecture by G.W. Gibbons.
