Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $

Luca Battaglia; Francesca Gladiali; Massimo Grossi

doi:10.3934/dcds.2020293

Article Contents

2021, Volume 41, Issue 2: 681-700. Doi: 10.3934/dcds.2020293

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$ L^2 $

-supercritical case

Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $

1.
Università degli Studi Roma Tre, Dipartimento di Matematica e Fisica, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy
2.
Università degli Studi di Sassari, Dipartimento di Chimica e Farmacia, Via Piandanna 4, 00710 Sassari, Italy
3.
Sapienza Università di Roma, Dipartimento di Matematica, Piazzale Aldo Moro 5, 00185 Roma, Italy

^* Corresponding author: Massimo Grossi
^* Corresponding author: Massimo Grossi

Received: February 29, 2020

Early access: August 2020

Published: February 2021

The third author is supported by "Fondi di ateneo Sapienza"

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Abstract

We consider the following Dirichlet problem

$\left\{ \begin{matrix} -\Delta u=\lambda f(u)\ \ \ \ \text{in}\ \Omega \\ u=0\ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial \Omega \\\end{matrix} \right.,\ \ \ \ \ \ \ \left( \mathcal{P}_{f}^{\lambda } \right)$

with $ \lambda<0 $ and $ f $ non-negative and non-decreasing.

We show existence and uniqueness of solutions $ u_\lambda $ for any $ \lambda $ and discuss their asymptotic behavior as $ \lambda\to-\infty $. In the expansion of $ u_\lambda $ large solutions naturally appear.

Keywords:

Mathematics Subject Classification: Primary:35B40;Secondary:35B44, 35A01, 35A02.

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