Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

中文翻译：

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局部与非局部椭圆方程：短程场相互作用

摘要 本文研究了一类结合局部和非局部算子的单参数椭圆方程族，即拉普拉斯算子和分数拉普拉斯算子。我们分析谱性质，建立最大值原理的有效性，证明弱解的存在、不存在、对称性和正则性结果。还研究了当耦合参数消失（将问题变成纯粹的非局部问题）或趋于无穷大（将问题简化为经典的半线性拉普拉斯方程）时弱解的渐近行为。