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Advanced Nonlinear Studies

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Volume 18, Issue 3

Issues

A Diffusion Equation with a Variable Reaction Order

Jorge García-Melián
  • Departamento de Análisis Matemático and IUEA, Universidad de La Laguna, P. O. Box 456, 38200 La Laguna, Tenerife, Spain
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/ Julio D. Rossi
  • Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pab. 1 (1428), Buenos Aires, Argentina
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/ José C. Sabina de Lis
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  • Departamento de Análisis Matemático and IUEA, Universidad de La Laguna, P. O. Box 456, 38200 La Laguna, Tenerife, Spain
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Published Online: 2017-09-02 | DOI: https://doi.org/10.1515/ans-2017-6030

Abstract

This paper deals with the problem

{-Δu=λuq(x),xΩ,u=0,xΩ,

where ΩN is a bounded smooth domain, λ>0 is a parameter and the reaction order q(x) is a Hölder continuous positive function satisfying q(x)>1 for all xΩ. The relevant feature here is that q is assumed to achieve the value one on Ω. By assuming that q is subcritical, our main result states the existence of a positive solution for all λ>0. We also study its asymptotic behavior as λ0 and as λ. It should be noticed that the fact that q=1 somewhere in Ω gives rise to serious difficulties when looking for critical points of the functional associated with the problem above. This work is a continuation of [13] where q is assumed to take values both greater and smaller than one in Ω, but is constrained to satisfy q(x)>1 on Ω.

Keywords: Variational Methods; Palais–Smale Sequences; A Priori Estimates

MSC 2010: 35J20; 35B45

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About the article


Received: 2017-06-09

Revised: 2017-08-16

Accepted: 2017-08-16

Published Online: 2017-09-02

Published in Print: 2018-08-01


Funding Source: Ministerio de Ciencia e Innovación

Award identifier / Grant number: MTM2014-52822-P

Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2014-52822-P

The authors were supported by Ministerio de Ciencia e Innovación and Ministerio de Economía y Competitividad under project MTM2014-52822-P.


Citation Information: Advanced Nonlinear Studies, Volume 18, Issue 3, Pages 555–566, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2017-6030.

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