A note on semilinear fractional elliptic equation: analysis and discretization^∗

¹ Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA.
hantil@gmu.edu
² Chair of Optimal Control, Center of Mathematical Sciences, Technical University of Munich, Boltzmannstraße 3, 85748 Garching by Munich, Germany.
pfefferer@ma.tum.de
³ University of Puerto Rico (Rio Piedras Campus), College of Natural Sciences, Department of Mathematics, PO Box 70377 San Juan PR 00936-8377 ( USA).
mahamadi.warma1@upr.edu, mjwarma@gmail.com

Received: 20 July 2016
Revised: 9 March 2017
Accepted: 20 April 2017

Abstract

In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order s ∈ (0,1). We identify minimal conditions on the nonlinear term and the source which lead to existence of weak solutions and uniform L^∞-bound on the solutions. Next we realize the fractional Laplacian as a Dirichlet-to-Neumann map via the Caffarelli−Silvestre extension. We introduce a first-degree tensor product finite elements space to approximate the truncated problem. We derive a priori error estimates and conclude with an illustrative numerical example.