Existence of positive solutions for a class of nonlinear elliptic equations with Hardy potential

Linlin Wang; Jingjing Liu; Yonghong Fan; Linlin Wang; Jingjing Liu; Yonghong Fan

doi:10.3934/math.2026140

AIMS Mathematics

2026, Volume 11, Issue 2: 3441-3463. doi: 10.3934/math.2026140

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Research article

Existence of positive solutions for a class of nonlinear elliptic equations with Hardy potential

School of Mathematics and Statistics Sciences, Ludong University, Yantai 264025, China

Received: 19 November 2025 Revised: 29 January 2026 Accepted: 30 January 2026 Published: 05 February 2026
MSC : 35B09, 35J60, 35J70

This paper is concerned with a class of nonlinear elliptic equations with a generalized Hardy potential
$ \begin{equation*} -\Delta u = \frac{\lambda }{|x{{|}^{\alpha }}}u-b(x)g(u), \ x\in \Omega \backslash \{0\}, \end{equation*} $
where $ \alpha > 0 $, $ \Omega \subset {\ \mathbb{R}}^{N}(N\geq 3) $ is a bounded smooth domain containing the origin. We establish the existence, nonexistence, and asymptotic behavior of positive solutions. When the potential function $ |x|{^{-\alpha }} $ has strong singularity at the origin, we obtain some qualitative properties of positive solutions.
- semilinear elliptic differential equation,
- upper and lower solutions,
- blow-up problem,
- uniqueness,
- asymptotic behavior
Citation: Linlin Wang, Jingjing Liu, Yonghong Fan. Existence of positive solutions for a class of nonlinear elliptic equations with Hardy potential[J]. AIMS Mathematics, 2026, 11(2): 3441-3463. doi: 10.3934/math.2026140

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Abstract

This paper is concerned with a class of nonlinear elliptic equations with a generalized Hardy potential

$ \begin{equation*} -\Delta u = \frac{\lambda }{|x{{|}^{\alpha }}}u-b(x)g(u), \ x\in \Omega \backslash \{0\}, \end{equation*} $

where $ \alpha > 0 $, $ \Omega \subset {\ \mathbb{R}}^{N}(N\geq 3) $ is a bounded smooth domain containing the origin. We establish the existence, nonexistence, and asymptotic behavior of positive solutions. When the potential function $ |x|{^{-\alpha }} $ has strong singularity at the origin, we obtain some qualitative properties of positive solutions.

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