Multiplicity of Solutions to Elliptic Problems Involving the 1-Laplacian with a Critical Gradient Term-Reference-Cited by-同舟云学术

Multiplicity of Solutions to Elliptic Problems Involving the 1-Laplacian with a Critical Gradient Term

Published:2017-04-21 Issue:2 Volume:17 Page:333-353
ISSN:1536-1365
Container-title:Advanced Nonlinear Studies
language:en
Short-container-title:

Author:

Abdellaoui Boumediene¹,Dall’Aglio Andrea²,Segura de León Sergio³

Affiliation:

1. Département de Mathématiques, Université Abou Bakr Belkaïd, Tlemcen, Tlemcen 13000, Algeria

2. Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale A. Moro 5, 00185 Roma, Italy

3. Departament d’Anàlisi Matemàtica, Universitat de València, Dr. Moliner 50, 46100 Burjassot, Valencia, Spain

Abstract

Abstract In the present paper we study the Dirichlet problem for an equation involving the 1-Laplacian and a total variation term as reaction. We prove a strong multiplicity result. Namely, we show that for any positive Radon measure concentrated in a set away from the boundary and singular with respect to a certain capacity, there exists an unbounded solution, and measures supported on disjoint sets generate different solutions. These results can be viewed as the analogue for the 1-Laplacian operator of some known multiplicity results which were first obtained by Ireneo Peral, to whom this article is dedicated, and his collaborators.

Funder

Ministerio de Economía y Competitividad

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics,Statistical and Nonlinear Physics

Link

https://www.degruyter.com/document/doi/10.1515/ans-2017-0011/pdf

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