On Singular Equations Involving Fractional Laplacian
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
ON SINGULAR EQUATIONS INVOLVING FRACTIONAL LAPLACIAN∗ Ahmed YOUSSFI† National School of Applied Sciences Sidi Mohamed Ben Abdellah University-Fez Laboratory of Mathematical Analysis and Applications My Abdellah Avenue, Road Imouzer, P.O. Box 72 F`es-Principale, 30 000, Fez, Morocco E-mail : [email protected]; [email protected]
Ghoulam OULD MOHAMED MAHMOUD National School of Applied Sciences Sidi Mohamed Ben Abdellah University-Fez My Abdellah Avenue, Road Imouzer, P.O. Box 72 F`es-Principale, 30 000, Fez, Morocco E-mail : [email protected] Abstract We study the existence and the regularity of solutions for a class of nonlocal equations involving the fractional Laplacian operator with singular nonlinearity and Radon measure data. Key words
fractional Laplacian; singular elliptic equations; measure data
2010 MR Subject Classification
1
35R11; 35J62; 46E35; 35A15
Introduction
Lately, problems involving nonlocal operators and singular terms have recently received considerable attention in the literature. A good amount of investigations have focused on the existence and/or regularity of solutions to such problems governed by the fractional Laplacian with a singularity due to a negative power of the unknown or described by a potential, see for instance, [1, 2, 4, 7, 8, 12] and related papers. A prototype of nonlocal operators is the fractional Laplacian operator of the form (−∆)s , 0 < s < 1, which is actually the infinitesimal generator of the radially symmetric and sstable L´evy processes [6]. Fractional Laplacian operators naturally arise from a wide range of applications. They appear, for instance, in thin obstacle problems [14], crystal dislocation [18], phase transition [30] and others. In this paper, we are interested in the existence and regularity of solutions to the following ∗ Received
February 18, 2019; revised September 7, 2019. author: Ahmed YOUSSFI.
† Corresponding
1290
ACTA MATHEMATICA SCIENTIA
Dirichlet problem
f (x) (−∆)s u = γ + µ in Ω, u u>0 u = 0
in Ω,
Vol.40 Ser.B
(1.1)
on RN \Ω,
where Ω is an open bounded subset in RN , N > 2s, of class C 0,1 , s ∈ (0, 1), γ > 0, f is a non-negative function on Ω, µ is a non-negative bounded Radon measure on Ω and (−∆)s is the fractional Laplacian operator of order 2s defined by Z u(x) − u(y) (−∆)s u = α(N, s)P.V. dy, N +2s N R |x − y| where “P.V.” stands for the integral in the principal value sense and α(N, s) is a positive renormalizing constant, depending only on N and s, given by α(N, s) = s
−1
4s Γ( N2 + s) π
N 2
s Γ(1 − s)
2s
so that the identity (−∆) u = F (|ξ| F u), ξ ∈ RN , s ∈ (0, 1) and u ∈ S(RN ) holds, where F u stands for the Fourier transform of u belonging to the Schwartz class S(RN ) (cf. [23]). More details on the operator (−∆)s and the asymptotic behaviour of α(N, s) can be found in [17]. The case s = 1 corresponds to the classical Laplacian operator. If further µ = 0
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