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ON NONLOCAL NONLINEAR ELLIPTIC PROBLEMS WITH THE FRACTIONAL LAPLACIAN

Part of: Variational problems in infinite-dimensional spaces Partial differential equations Elliptic equations and systems Global differential geometry

Published online by Cambridge University Press:  29 January 2019

LI MA
LI MA*
Affiliation:
School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing 100083, China; Department of Mathematics, Henan Normal University, Xinxiang 453007, China e-mail: [email protected]
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Abstract

In this paper, we study the existence of positive solutions to a semilinear nonlocal elliptic problem with the fractional α-Laplacian on Rn, 0 < α < n. We show that the problem has infinitely many positive solutions in $ {C^\tau}({R^n})\bigcap H_{loc}^{\alpha /2}({R^n}) $. Moreover, each of these solutions tends to some positive constant limit at infinity. We can extend our previous result about sub-elliptic problem to the nonlocal problem on Rn. We also show for α ∊ (0, 2) that in some cases, by the use of Hardy’s inequality, there is a nontrivial non-negative $ H_{loc}^{\alpha /2}({R^n}) $ weak solution to the problem

$$ {( - \Delta )^{\alpha /2}}u(x) = K(x){u^p} \quad {\rm{ in}} \ {R^n}, $$
where K(x) = K(|x|) is a non-negative non-increasing continuous radial function in Rn and p > 1.

MSC classification

Secondary: 35A15: Variational methods 35B50: Maximum principles 35J60: Nonlinear elliptic equations 53C70: Direct methods ($G$-spaces of Busemann, etc.) 58E50: Applications
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 1 , January 2020 , pp. 75 - 84
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019 

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