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Estimate, existence and nonexistence of positive solutions of Hardy–Hénon equations

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  18 May 2021

Xiyou Cheng ,
Lei Wei  and
Yimin Zhang
Xiyou Cheng
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu730000, China ([email protected])
Lei Wei
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu221116, China ([email protected])
Yimin Zhang
Affiliation:
Center for Mathematical Sciences and Department of Mathematics, Wuhan University of Technology, Wuhan430070, China ([email protected])
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Abstract

We consider the boundary Hardy–Hénon equation

\[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \]
where $B_1(0)\subset \mathbb {R}^{N}$$(N\geq 3)$ is a ball of radial $1$ centred at $0$, $p>0$ and $\alpha \in \mathbb {R}$. We are concerned with the estimate, existence and nonexistence of positive solutions of the equation, in particular, the equation with Dirichlet boundary condition. For the case $0< p<({N+2})/({N-2})$, we establish the estimate of positive solutions. When $\alpha \leq -2$ and $p>1$, we give some conclusions with respect to nonexistence. When $\alpha >-2$ and $1< p<({N+2})/({N-2})$, we obtain the existence of positive solution for the corresponding Dirichlet problem. When $0< p\leq 1$ and $\alpha \leq -2$, we show the nonexistence of positive solutions. When $0< p<1$, $\alpha >-2$, we give some results with respect to existence and uniqueness of positive solutions.

Keywords

SingularityEstimateExistenceNonexistence

MSC classification

Primary: 35J61: Semilinear elliptic equations
Secondary: 35A09: Classical solutions 35B09: Positive solutions 35B45: A priori estimates
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 518 - 541
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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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