Hostname: page-component-5b777bbd6c-gtgcz Total loading time: 0 Render date: 2025-06-22T19:50:58.896Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Ambrosetti, A. and Rabinowitz, P.. Dual variational methods in critical points theory and applications. J. Funct. Anal. 14 (1973), 349–381.CrossRefGoogle Scholar
2Ambrosetti, A., Brezis, H. and Cerami, G.. Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122 (1994), 519–543.CrossRefGoogle Scholar
3Brezis, H. and Nirenberg, L.. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36 (1983), 437–447.CrossRefGoogle Scholar
4de Figueiredo, D., Gossez, J.-P. and Ubilla, P.. Local superlinearity and sublinearity for indefinite semilinear elliptic problems. J. Funct. Anal. 199 (2003), 452–467.CrossRefGoogle Scholar
5de Figueiredo, D., Gossez, J.-P. and Ubilla, P.. Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity. J. Eur. Math. Soc. 8 (2006), 269–286.CrossRefGoogle Scholar
6do Ó, J. M., Ruf, B. and Ubilla, P.. On supercritical Sobolev type inequalities and related elliptic equations. Calc. Var. Partial Differ. Equ. 55 (2016), 18, Article 83CrossRefGoogle Scholar
7Pohozaev, S.. Eigenfunctions of the equation Δ u + λ f(u). Sov. Math. Dokl. 6 (1965), 1408–1411.Google Scholar
8Strauss, W.. Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55 (1977), 149–162.CrossRefGoogle Scholar