Exact singular behavior of positive solutions to nonlinear elliptic equations with a Hardy potential

Abstract

In this paper, we study the singular behavior at x=0of positive solutions to the equation −u = λ |x|2 u − |x|σ up, x ∈ \{0}, where ⊂RN(N≥3)is a bounded domain with 0 ∈, and p>1, σ>−2are given constants. For the case λ ≤(N−2)2/4, the singular behavior of all the positive solutions is completely classified in the recent paper [5]. Here we determine the exact singular behavior of all the positive solutions for the remaining case λ >(N−2)2/4. In sharp contrast to the case λ ≤(N−2)2/4, where several converging/blow-up rates of u(x)are possible as |x| →0, we show that when λ >(N−2)2/4, every positive solution u(x)blows up in the same fashion: lim |x|→0 |x| 2+σ p−1 u(x) = λ+ 2 +σ p −1 2 +σ p − 1 +2 −N 1/(p−1) .