Special Issue in honor of John W. Neuberger. Electron. J. Diff. Eqns., Special Issue 02 (2023), pp. 135-149.

Positive solutions for nonlinear fractional Laplacian problems

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 a positive weak solution for classes of nonlinearities which are either sublinear or asymptotically linear at infinity. We use the method of sub-and-supersolutions to establish the results. We also provide numerical bifurcation diagrams, corresponding to the theoretical results, using the finite element method in one dimension.

Published March 27, 2023.
Math Subject Classifications: 35J60, 35J61, 35R11.
Key Words: Fractional Laplacian; sublinear; asymptotically linear; sub- and supersolution; positive weak solution.
DOI: https://doi.org/10.58997/ejde.sp.02.h1

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Elliott Hollifield
The University of North Carolina
Pembroke, NC 28372, USA
email: Elliott.Hollifield@uncp.edu

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