Local versus nonlocal elliptic equations: short-long range field interactions

Advances in Nonlinear Analysis

article

Daniele Cassani¹  Luca Vilasi¹  Youjun Wang² 

[1] and RISM-Riemann International School of Mathematics Villa Toeplitz;Department of Mathematics South China University of Technology Guangzhou;Dip. di Scienza e Alta Tecnologia Universitá degli Studi dell'Insubria

关键词: Nonlocal and local Laplacians;    asymptotic behavior;    variational methods;   

DOI  :  10.1515/anona-2020-0166

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

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.

【 授权许可】

CC BY