Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary. Issue 12 (2nd December 2017)

• Record Type: Journal Article
• Authors: Chen, Huyuan; Alhomedan, Suad; Hajaiej, Hichem; Markowich, Peter
• Is Part Of: Complex variables and elliptic equations. Volume 62:Issue 12(2017)
• Page Start: 1687
• Page End: 1729
• Publication Date: 2017-12-02
• Subjects: Fractional Laplacian -- Radon measure -- Dirac mass -- Green kernel -- Schauder's fixed point theorem; 35R11 -- 35J61 -- 35R06; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2016.1278441
• Languages: English
• ISSNs: 1747-6933

Abstract

Abstract : The purpose of this paper is to study the weak solutions of the fractional elliptic problem (Section.Display) where, or, with is the fractional Laplacian defined in the principle value sense, is a bounded open set in with, is a bounded Radon measure supported in and is defined in the distribution sense, i.e. here denotes the unit inward normal vector at . In this paper, we prove that (0.1) with admits a unique weak solution when g is a continuous nondecreasing function satisfying Our interest then is to analyse the properties of weak solution when with, including the asymptotic behaviour near and the limit of weak solutions as . Furthermore, we show the optimality of the critical value in a certain sense, by proving the non-existence of weak solutions when . The final part of this article is devoted to the study of existence for positive weak solutions to (0.1) when and is a bounded nonnegative Radon measure supported in . We employ the Schauder's fixed point theorem to obtain positive solution under the hypothesis that g is a continuous function satisfying -pagination