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

Published online by Cambridge University Press:  28 September 2021

Lucio Boccardo
Affiliation:
Istituto Lombardo and Sapienza Università di Roma, P.le Aldo Moro 5, 00185, Roma (boccardo@uniroma1.it)
Liliane Maia
Affiliation:
Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília, Brazil (lilimaia@unb.br)
Benedetta Pellacci
Affiliation:
Dipartimento di Matematica e Fisica, Università della Campania ‘Luigi Vanvitelli’, V.le A. Lincoln 5, Caserta 81100, Italy (benedetta.pellacci@unicampania.it)

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.

Type
Research Article
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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