Singular solutions of a Hénon equation involving a nonlinear gradient term

Craig Cowan; Abdolrahman Razani

doi:10.3934/cpaa.2021172

Article Contents

2022, Volume 21, Issue 1: 141-158. Doi: 10.3934/cpaa.2021172

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Singular solutions of a Hénon equation involving a nonlinear gradient term

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, 34149-16818, Qazvin, Iran

^* Corresponding author
^* Corresponding author

Received: February 28, 2021

Revised: August 31, 2021

Early access: September 2021

Published: January 2022

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Abstract

Here, we consider positive singular solutions of

$ \begin{equation*} \left\{ \begin{array}{lcc} -\Delta u = |x|^\alpha |\nabla u|^p & \text{in}& \Omega \backslash\{0\},\\ u = 0&\text{on}& \partial \Omega, \end{array} \right. \end{equation*} $

where $ \Omega $ is a small smooth perturbation of the unit ball in $ \mathbb{R}^N $ and $ \alpha $ and $ p $ are parameters in a certain range. Using an explicit solution on $ B_1 $ and a linearization argument, we obtain positive singular solutions on perturbations of the unit ball.

Keywords:

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

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

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