Positive solutions of elliptic equations with a strong singular potential

Abstract

<p>In this paper, we study positive solutions of the elliptic equation </p><p> −Δ𝑢=𝜆𝑑(𝑥)𝛼𝑢−𝑑(𝑥)𝜎𝑢𝑝inΩ, </p><p> 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 </p><p> lim𝑑(𝑥)→0𝑢(𝑥)𝑑(𝑥)𝛼+𝜎𝑝−1=𝜆1𝑝−1.</p>