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On supercritical problems involving the Laplace operator

Published online by Cambridge University Press:  27 February 2020

Rodrigo Clemente
Affiliation:
Department of Mathematics, Universidade Federal Rural de Pernambuco, 52171-900Recife, Pernambuco, Brazil ([email protected])
João Marcos do Ó*
Affiliation:
Department of Mathematics, Federal University of Paraíba, 58051-900 João Pessoa, Paraíba, Brazil ([email protected])
Pedro Ubilla
Affiliation:
Departamento de Matemáticas y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile ([email protected])
*
*Corresponding author.

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.

Type
Research Article
Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

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