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Research Article | Open Access

Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary

S. Bandyopadhyay1, M. Chhetri1( ), B. B. Delgado2, N. Mavinga3, R. Pardo4
UNC Greensboro, Greensboro, NC 27402, USA
Universidad Autónoma de Aguascalientes, 20131 Aguascalientes, Mexico
Swarthmore College, Swarthmore, PA 19081, USA
Universidad Complutense de Madrid, 28040 Madrid, Spain
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Abstract

This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.

Keywords

elliptic problemnonlinear boundary conditionsmaximal and minimal weak solutionpseudomonotone operatorKato's inequality

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Electronic Research Archive
Volume 30 Issue 6,
June 2022
Pages 2121-2137
DOI: 10.3934/era.2022107

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Bandyopadhyay S, Chhetri M, Delgado BB, et al. Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary. Electronic Research Archive, 2022, 30(6): 2121-2137. https://doi.org/10.3934/era.2022107

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Received: 14 October 2021
Revised: 29 December 2021
Accepted: 16 January 2022
Published: 15 June 2022
©2022 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)