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On supercritical problems involving the Laplace operator

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  27 February 2020

Rodrigo Clemente ,
João Marcos do Ó  and
Pedro Ubilla
Rodrigo Clemente
Affiliation:
Department of Mathematics, Universidade Federal Rural de Pernambuco, 52171-900Recife, Pernambuco, Brazil (rodrigo.clemente@ufrpe.br)
João Marcos do Ó*
Affiliation:
Department of Mathematics, Federal University of Paraíba, 58051-900 João Pessoa, Paraíba, Brazil (jmbo@pq.cnpq.br)
Pedro Ubilla
Affiliation:
Departamento de Matemáticas y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile (pubilla@usach.cl)
*
*Corresponding author.
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Abstract

We discuss the existence, nonexistence and multiplicity of solutions for a class of elliptic equations in the unit ball with zero Dirichlet boundary conditions involving nonlinearities with supercritical growth. By using Pohozaev type identity we prove a nonexistence result for a class of supercritical problems with variable exponent which allow us to complement the analysis developed in (Calc. Var. (2016) 55:83). Moreover, we establish existence results of positive solutions for semilinear elliptic equations involving nonlinearities which are subcritical at infinity just in a part of the domain, and can be supercritical in a suitable sense.

Keywords

Critical Sobolev exponentSupercritical elliptic problemsRadial solutions

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J25: Boundary value problems for second-order elliptic equations 35J50: Variational methods for elliptic systems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 187 - 201
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Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

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References

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