Electron. J. Differential Equations, Vol. 2020 (2020), No. 81, pp. 1-31.

Existence of positive solutions for fractional Laplacian equations: theory and numerical experiments

Maya Chhetri, Petr Girg, Elliott Hollifield

Abstract:
We consider a class of nonlinear fractional Laplacian problems satisfying the homogeneous Dirichlet condition on the exterior of a bounded domain. We prove the existence of positive weak solution for classes of sublinear nonlinearities including logistic type. A method of sub- and supersolution, without monotone iteration, is established to prove our existence results. We also provide numerical bifurcation diagrams and the profile of positive solutions, corresponding to the theoretical results using the finite element method in one dimension.

Submitted June 28, 2019. Published July 28, 2020.
Math Subject Classifications: 35J60, 35J61, 35R11.
Key Words: Fractional Laplacian; sub- and supersolution; sublinear; logistic equation; finite element method.
DOI: 10.58997/ejde.2020.81

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Maya Chhetri
Department of Mathematics and Statistics
The University of North Carolina at Greensboro, NC 27402, USA
email: maya@uncg.edu
Petr Girg
Department of Mathematics
Faculty of Applied Sciences
University of West Bohemia
Univerzitni 8, 30100 Plzen, Czech Republic
email: pgirg@kma.zcu.cz
Elliott Hollifield Department of Mathematics and Statistics
The University of North Carolina at Greensboro, NC 27402 USA
email: ezhollif@uncg.edu

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