带有Hardy和Sobolev-Hardy临界指标项的扰动椭圆方程的解

Positive Solutions of Perturbation Elliptic Equation Involving Hardy Potential and Critical Sobolev-Hardy Exponent

考虑如下带有Hardy和Sobolev-Hardy临界指标项的扰动椭圆方程这里2*(s)=(2(N-s))/(N-2)是Sobolev-Hardy临界指标,N≥3,λ∈R,0≤s<2,1<q<2*-1,0≤μ<u=((N-2)~2)/4,a(x)∈C(R^N).在|λ|足够小的情况下,应用临界点理论中的扰动方法来得到方程(0.1)正解的存在性.接下来考虑anisotropic椭圆方程b(x)∈C(R^N).在|λ|足够小的情况下,应用临界点理论中的扰动方法来得到方程(0.2)正解的存在性.

In this paper, we are concerned with the following elliptic equation involving critical Sobolev-Hardy exponent {-△u-μu/｜x｜^2＋λa（x）u^q=｜u｜^2＊（s）^-2/｜x｜^su,x∈R^N,u〉0,u∈D^1,2（R^N）, where 2^＊（s）=（2（N-s））/（N-2） is the critical Sobolev-Hardy exponent,N≥3,λ∈R,0≤s〈2,1〈q〈2^＊-1,0≤μ〈μ^-=（N-2）^2/4,a（x）∈C（R^N）We firstly use an abstract perturbation method in critical point theory to obtain the existence results of positive solutions of the equation for small value of ｜λ｜. Secondly, we focus on an anisotropic elliptic equation of the form {-div[（1＋λb（x））△u]＋λa（x）u^q=μu/｜x｜^2＋｜u｜^2＊（s）-2/｜x｜^2u,x∈R^N, u〉0,u∈D^1,2（R^N）, The same abstract method is used to yield existence result of positive solutions of the equation for small value of ｜λ｜.