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

Authors
  • Cassani, Daniele1
  • Vilasi, Luca1
  • Wang, Youjun2, 3
  • 1 Dip. di Scienza e Alta Tecnologia, Universitá degli Studi dell'Insubria, and RISM-Riemann International School of Mathematics Villa Toeplitz, Via G.B. Vico , (Italy)
  • 2 Department of Mathematics South China University of Technology Guangzhou, P. R. China , (China)
  • 3 Dip. di Scienza e Alta Tecnologia Universitá degli Studi dell'Insubria via Valleggio, Italy , (Italy)
Type
Published Article
Journal
Advances in Nonlinear Analysis
Publisher
De Gruyter
Publication Date
Dec 31, 2020
Volume
10
Issue
1
Pages
895–921
Identifiers
DOI: 10.1515/anona-2020-0166
Source
De Gruyter
Keywords
License
Green

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.

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