Positive solutions of elliptic equations with a strong singular potential

2019

Wei, Lei

Du, Yihong

Journal Article

en

Publication

Wiley-Blackwell Publishing Ltd

United Kingdom

10.1112/blms.12229

une:1959.11/28572

In this paper, we study positive solutions of the elliptic equation

−Δ𝑢=𝜆𝑑(𝑥)𝛼𝑢−𝑑(𝑥)𝜎𝑢𝑝inΩ,

where 𝛼>2,𝜎>−𝛼, 𝑝>1, 𝑑(𝑥)=𝑑𝑖𝑠𝑡(𝑥,𝜕Ω), and Ω is a bounded smooth domain in ℝ𝑁(𝑁⩾2). When 𝛼=2, the term 1𝑑(𝑥)𝛼=1𝑑(𝑥)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 𝜕Ω. We show that when 𝜆<0, the equation has no positive solution, while when 𝜆>0, the equation has a unique positive solution, and it satisfies

lim𝑑(𝑥)→0𝑢(𝑥)𝑑(𝑥)𝛼+𝜎𝑝−1=𝜆1𝑝−1.

link

Bulletin of the London Mathematical Society, 51(2), p. 251-266

1469-2120

0024-6093

251

266