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Asymptotic behaviour of positive solutions of semilinear elliptic problems with increasing powers

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  28 September 2021

Lucio Boccardo ,
Liliane Maia Open the ORCID record for Liliane Maia [Opens in a new window]  and
Benedetta Pellacci Open the ORCID record for Benedetta Pellacci [Opens in a new window]
Lucio Boccardo
Affiliation:
Istituto Lombardo and Sapienza Università di Roma, P.le Aldo Moro 5, 00185, Roma ([email protected])
Liliane Maia
Affiliation:
Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília, Brazil ([email protected])
Benedetta Pellacci
Affiliation:
Dipartimento di Matematica e Fisica, Università della Campania ‘Luigi Vanvitelli’, V.le A. Lincoln 5, Caserta 81100, Italy ([email protected])
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Abstract

We prove existence results of two solutions of the problem

\[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{in}\ \Omega,\\ u>0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial \Omega, \end{cases} \]
where $L(v)=-\textrm {div}(M(x)\nabla v)$ is a linear operator, $p\in (2,2^{*}]$ and $\lambda$ and $m$ sufficiently large. Then their asymptotical limit as $m\to +\infty$ is investigated showing different behaviours.

Keywords

Increasing powers in the lower order termAsymptotical analysisSemilinear elliptic equations

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35J66: Nonlinear boundary value problems for nonlinear elliptic equations 35J20: Variational methods for second-order elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 5 , October 2022 , pp. 1233 - 1250
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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