Positive solutions of elliptic equations with a strong singular potential

Abstract

In this paper, we study positive solutions of the elliptic equation − Δ u = λ d ( x ) α u − d ( x ) σ u p in Ω , where α > 2 , σ > − α , p > 1 , d ( x ) = d i s t ( x , ∂ Ω ) and Ω is a bounded smooth domain in R N ( N ⩾ 2 ) . When α = 2 , the term 1 d ( x ) α = 1 d ( 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 lim d ( x ) → 0 u ( x ) d ( x ) α + σ p − 1 = λ 1 p − 1 .