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 − Δ + ( − Δ)s, with s ∈ (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.

中文翻译：

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涉及混合局部和非局部算子的半线性椭圆方程

在本文中，我们将椭圆算子视为经典二阶微分算子和分数型非局部算子的叠加。尽管我们开发的方法非常通用，但为了具体起见，我们关注运算符采用形式的情况 - Δ + ( - Δ)s， 和s∈ (0, 1)。我们在这里关注解的对称性，我们证明了基于移动平面方法的径向对称结果，以及与 GW Gibbons 的经典猜想相关的一维对称结果。