We present a new method to determine the optimal regularity of positive solutions u∈C^(4)(Ω\{0})∩C^(0)(Ω) of the Hénon-Hardy equation,i.e.,Δ^(2)u=|x|^(α)u^(p)inΩ,(0,1) where Ω■RN(N≥4) is a bounded smoot...We present a new method to determine the optimal regularity of positive solutions u∈C^(4)(Ω\{0})∩C^(0)(Ω) of the Hénon-Hardy equation,i.e.,Δ^(2)u=|x|^(α)u^(p)inΩ,(0,1) where Ω■RN(N≥4) is a bounded smooth domain with 0∈Ω,α>-4,and p∈R.It is clear that 0 is an isolated singular point of solutions of(0.1) and the optimal regularity of u in Ω relies on the parameter α.It is also important to see that the regularity of u at x=0 determines the regularity of u in Ω.We first establish asymptotic expansions up to arbitrary orders at x=0 of prescribed positive solutions u ∈C^(4)(Ω{0}) ∩ C^(0)(Ω)of(0.1).Then we show that the regularity at x=0 of each positive solution u of(0.1) can be determined by some terms in asymptotic expansions of the related positive radial solution of the equation(0.1) with Ω=B,where B is the unit ball of R^(N).The main idea works for more general equations with singular weights.展开更多
To understand genetic variation and population dispersal in the Yangtze vole Microtusfortis calamorum distributed in the Dongting Lake region, 144 individuals were collected from six habitat patches. The mitochondrial...To understand genetic variation and population dispersal in the Yangtze vole Microtusfortis calamorum distributed in the Dongting Lake region, 144 individuals were collected from six habitat patches. The mitochondrial DNA control region was sequenced and 17 haplotypes were observed. Of the six investigated populations, haplotype and nucleotide diversities of those from larger patches were higher than those from smaller patches. Nonparametric correlation analysis showed that patch size had a positive correlation with haplotype diversity （r = 0.943, P 〈 0.01）. A neighbour-joining tree of the 17 haplotypes showed no geo- graphic genetic structure among the six populations. Analysis of isolation by distance showed that genetic differentiation among the six populations was not positively related to geographic distance. Analysis of mismatch distribution indicated that the voles had passed through a population expansion. The pattern of haplotype distribution in the Changsha population suggests that the population was established by a founder effect展开更多
基金supported by National Natural Science Foundation of China (Grant No. 11571093)supported by the Fundamental Research Funds for the Central Universities (Grant No. WK0010000064)Anhui Provincial Natural Science Foundation (Grant No. BJ0010000026)。
文摘We present a new method to determine the optimal regularity of positive solutions u∈C^(4)(Ω\{0})∩C^(0)(Ω) of the Hénon-Hardy equation,i.e.,Δ^(2)u=|x|^(α)u^(p)inΩ,(0,1) where Ω■RN(N≥4) is a bounded smooth domain with 0∈Ω,α>-4,and p∈R.It is clear that 0 is an isolated singular point of solutions of(0.1) and the optimal regularity of u in Ω relies on the parameter α.It is also important to see that the regularity of u at x=0 determines the regularity of u in Ω.We first establish asymptotic expansions up to arbitrary orders at x=0 of prescribed positive solutions u ∈C^(4)(Ω{0}) ∩ C^(0)(Ω)of(0.1).Then we show that the regularity at x=0 of each positive solution u of(0.1) can be determined by some terms in asymptotic expansions of the related positive radial solution of the equation(0.1) with Ω=B,where B is the unit ball of R^(N).The main idea works for more general equations with singular weights.
文摘To understand genetic variation and population dispersal in the Yangtze vole Microtusfortis calamorum distributed in the Dongting Lake region, 144 individuals were collected from six habitat patches. The mitochondrial DNA control region was sequenced and 17 haplotypes were observed. Of the six investigated populations, haplotype and nucleotide diversities of those from larger patches were higher than those from smaller patches. Nonparametric correlation analysis showed that patch size had a positive correlation with haplotype diversity （r = 0.943, P 〈 0.01）. A neighbour-joining tree of the 17 haplotypes showed no geo- graphic genetic structure among the six populations. Analysis of isolation by distance showed that genetic differentiation among the six populations was not positively related to geographic distance. Analysis of mismatch distribution indicated that the voles had passed through a population expansion. The pattern of haplotype distribution in the Changsha population suggests that the population was established by a founder effect
