Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

Huyuan Chen; Suad Alhomedan; Hichem Hajaiej; Peter A. Markowich

doi:10.1080/17476933.2016.1278441

Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

Huyuan Chen, Suad Alhomedan, Hichem Hajaiej, Peter A. Markowich

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Abstract

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

Original language	English (US)
Pages (from-to)	1687-1729
Number of pages	43
Journal	Complex Variables and Elliptic Equations
Volume	62
Issue number	12
DOIs	https://doi.org/10.1080/17476933.2016.1278441
State	Published - Feb 6 2017

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: Suad Alhemedan extends his appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group NO (RGP- RG-1438-047). H. Chen is supported by NNSF of China [grant number 11401270], [grant number 11661045]; Jiangxi Provincial Natural Science Foundation [grant number 20161ACB20007].

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10.1080/17476933.2016.1278441

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title = "Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary",

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

author = "Huyuan Chen and Suad Alhomedan and Hichem Hajaiej and Markowich, {Peter A.}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: Suad Alhemedan extends his appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group NO (RGP- RG-1438-047). H. Chen is supported by NNSF of China [grant number 11401270], [grant number 11661045]; Jiangxi Provincial Natural Science Foundation [grant number 20161ACB20007].",

year = "2017",

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doi = "10.1080/17476933.2016.1278441",

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T1 - Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

AU - Chen, Huyuan

AU - Alhomedan, Suad

AU - Hajaiej, Hichem

AU - Markowich, Peter A.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: Suad Alhemedan extends his appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group NO (RGP- RG-1438-047). H. Chen is supported by NNSF of China [grant number 11401270], [grant number 11661045]; Jiangxi Provincial Natural Science Foundation [grant number 20161ACB20007].

PY - 2017/2/6

Y1 - 2017/2/6

N2 - The purpose of this paper is to study the weak solutions of the fractional elliptic problem(Formula presented.) where (Formula presented.), (Formula presented.) or (Formula presented.), (Formula presented.) with (Formula presented.) is the fractional Laplacian defined in the principle value sense, (Formula presented.) is a bounded (Formula presented.) open set in (Formula presented.) with (Formula presented.), (Formula presented.) is a bounded Radon measure supported in (Formula presented.) and (Formula presented.) is defined in the distribution sense, i.e.(Formula presented.) here (Formula presented.) denotes the unit inward normal vector at (Formula presented.). In this paper, we prove that (0.1) with (Formula presented.) admits a unique weak solution when g is a continuous nondecreasing function satisfying(Formula presented.) Our interest then is to analyse the properties of weak solution when (Formula presented.) with (Formula presented.), including the asymptotic behaviour near (Formula presented.) and the limit of weak solutions as (Formula presented.). Furthermore, we show the optimality of the critical value (Formula presented.) in a certain sense, by proving the non-existence of weak solutions when (Formula presented.). The final part of this article is devoted to the study of existence for positive weak solutions to (0.1) when (Formula presented.) and (Formula presented.) is a bounded nonnegative Radon measure supported in (Formula presented.). We employ the Schauder’s fixed point theorem to obtain positive solution under the hypothesis that g is a continuous function satisfying(Formula presented.)-pagination

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

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UR - http://www.tandfonline.com/doi/full/10.1080/17476933.2016.1278441

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DO - 10.1080/17476933.2016.1278441

M3 - Article

SN - 1747-6933

VL - 62

SP - 1687

EP - 1729

JO - Complex Variables and Elliptic Equations

JF - Complex Variables and Elliptic Equations

IS - 12

ER -

Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

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