关键词: 扰动方程, Sobolev-Hardy临界指标, 正解

Abstract:

In this paper, we are concerned with the following elliptic equation involving critical Sobolev-Hardy exponent
-Δu-μ(u)/(|x|²)+λa(x)u^q=(|u|^2*(s)-2)/(|x|^s)u,x∈R^N,(0.1)
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=μ(u)/(|x|²)+(|u|^2*(s)-2)/(|x|^s)u,x∈R^N,(0.2)
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|.

Key words: Perturbed, Critical Sobolev-Hardy Exponent, Positive Solution