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

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• 作者：Lei Wei^a ; ^{wlxznu@163.com} ; Yihong Du^b ; ^{ydu@turing.une.edu.au}
• 关键词：35J55 ; 35B40
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 April 2017
• 年：2017
• 卷：262
• 期：7
• 页码：3864-3886
• 全文大小：319 K
• 卷排序：262

文摘

In this paper, we study the singular behavior at x=0x=0 of positive solutions to the equation−Δu=λ|x|2u−|x|σup,x∈Ω\{0}, where Ω⊂RN(N≥3)Ω⊂RN(N≥3) is a bounded domain with 0∈Ω0∈Ω, and p>1p>1, σ>−2σ>−2 are given constants. For the case λ≤(N−2)2/4λ≤(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)2/4λ>(N−2)2/4. In sharp contrast to the case λ≤(N−2)2/4λ≤(N−2)2/4, where several converging/blow-up rates of u(x)u(x) are possible as |x|→0|x|→0, we show that when λ>(N−2)2/4λ>(N−2)2/4, every positive solution u(x)u(x) blows up in the same fashion:lim|x|→0⁡|x|2+σp−1u(x)=[λ+2+σp−1(2+σp−1+2−N)]1/(p−1).