Positive solutions of elliptic equations with a strong singular potential

Author(s)
Wei, Lei
Du, Yihong
Publication Date
2019
Abstract
<p>In this paper, we study positive solutions of the elliptic equation </p><p> βˆ’Ξ”π‘’=πœ†π‘‘(π‘₯)π›Όπ‘’βˆ’π‘‘(π‘₯)πœŽπ‘’π‘inΞ©, </p><p> where 𝛼>2,𝜎>βˆ’π›Ό, 𝑝>1, 𝑑(π‘₯)=𝑑𝑖𝑠𝑑(π‘₯,πœ•Ξ©), and Ξ© is a bounded smooth domain in ℝ𝑁(𝑁⩾2). When 𝛼=2, the term 1𝑑(π‘₯)𝛼=1𝑑(π‘₯)2 is often called a Hardy potential, and the equation in this case has been extensively investigated. Here we consider the case 𝛼>2, which gives a stronger singularity than the Hardy potential near πœ•Ξ©. We show that when πœ†<0, the equation has no positive solution, while when πœ†>0, the equation has a unique positive solution, and it satisfies </p><p> lim𝑑(π‘₯)β†’0𝑒(π‘₯)𝑑(π‘₯)𝛼+πœŽπ‘βˆ’1=πœ†1π‘βˆ’1.</p>
Citation
Bulletin of the London Mathematical Society, 51(2), p. 251-266
ISSN
1469-2120
0024-6093
Link
Publisher
Wiley-Blackwell Publishing Ltd
Title
Positive solutions of elliptic equations with a strong singular potential
Type of document
Journal Article
Entity Type
Publication

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