奇异摄动与临界的多调和方程

令整数k≥1,k＊=2N/（N-2k）（N≥2k＋1）。本文用变分方法首先证明了方程（-△）^ku=｜u｜^k＊-2 u＋λf（x）u,x∈Ω当Ω关于0点是一星型区域且f（x）=1/｜x｜^2k时没有非零解；其次证明了若f（x）〉0,f（x）∈Lloc^∞（Ω／{0}）且满足（1）存在β满足max{0,4k-N}≤β〈2k使得0〈lim｜x｜→0｜x｜^β f（x）=c〈∞；（2）存在δ〉μk使得对α.e.x∈Ω有｜x｜^β f（x）≤1/λδ，则P（k,f）在H0^k（Ω）中有一个非零解。

Singular Perturbation and Critical Polyharmonic Equation

Let k≥1 be an integer and k＊=2N/（N-2k）（N≥2k＋1）.In this paper, by using variational methods,we firstly prove that （-△）^ku=｜u｜^k＊-2 u＋λf（x）u,x∈Ω does not possess nontrivial solutions provided Ω is star shape with respect to 0 and f（x）=1/｜x｜^2k .Secondly, we prove that if f（x）〉0,f（x）∈Lloc^∞（Ω／{0}） and staisfies （ 1 ） β satisfying max{0,4k - N}≤ β 〈 2k such that 0 〈 lim｜x｜→0｜x｜^β f（x） = c 〈 ∞ ; （2） δ 〉 μk such that for α.e.x∈Ω,｜x｜^β f（x）≤1/λδ,then P（k, f） has a nontrivial solution in H0^k（Ω）.