Singular solutions of a Lane-Emden system

Craig Cowan; Abdolrahman Razani

doi:10.3934/dcds.2020291

Article Contents

2021, Volume 41, Issue 2: 621-656. Doi: 10.3934/dcds.2020291

This issue Previous Article Maximal factors of order

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of dynamical cubespaces Next Article Uniform stability estimate for the Vlasov-Poisson-Boltzmann system

Singular solutions of a Lane-Emden system

Craig Cowan^1, and
Abdolrahman Razani^2, ,

1.
Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
2.
Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Postal Code: 34149-16818, Qazvin, Iran

^* Corresponding author: A. Razani
^* Corresponding author: A. Razani

This work was done when the second author visited the Department of Mathematics, University of Manitoba on a sabbatical leave from Imam Khomeini International University. He appreciates the Department of Mathematics, University of Manitoba, for the hospitality and Prof. C. Cowan for his support.

Received: November 2019

Revised: May 2020

Early access: August 2020

Published: February 2021

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Abstract

In this work we consider the existence of positive singular solutions

$ \begin{equation} \left\{ \begin{array}{lcl} \hfill -\Delta u_1 & = & \lambda_1 | \nabla u_2|^p \qquad \mbox{ in } \Omega, \\ \hfill -\Delta u_2 & = & \lambda_2 | \nabla u_1|^q \qquad \mbox{ in } \Omega, \\ \hfill u_1 = u_2 & = & 0 \hfill \mbox{ on } \partial \Omega, \end{array}\right.\;\;\;\;\;\;\;(1) \end{equation} $

where $ \Omega $ is small $ C^2 $ perturbation of the unit ball $ B_1 $ in $ \mathbb{R}^N $ and $ \lambda_i $ are positive constants. Under suitable conditions on $ p $ and $ q $ we prove the existence of positive singular solutions of (1). We also examine the case where one or both of $ u_1,u_2 $ are Hölder continuous.

Keywords:

Lane-Emden equation,
singular solutions,
Laplace equation involving the gradient term,
semilinear elliptic equations,
perturbations of nonlinear operators.

Mathematics Subject Classification: Primary:34B18;Secondary:35J61, 35J75, 35J91, 35D99, 47H14.

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