关键词: Critical Sobolev-Hardy exponent, elliptic equation, Mountain Pass theorem, subsuper solutions, nonhomogeneous

Abstract:

For the following elliptic problem
-△ u - μu/|x|²={|u|^2*(s)-2u}/|x|^s+h(x), on R^N
u ∈ D^1,2(R^N), N≥3, 0≤μ<\bar\mu=(N-2)^{2 _/}4, 0≤ s<2,
where 2*(s)=2(N-s)/(N-2) is the critical Sobolev-Hardy exponent, h(x)∈ D^1,2(R^N))* , the dual space of (D^1,2(R^N)), with h(x)≥(≠) 0. By Ekeland's variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if
||h||_*N,s A_s ^(n-s)/(4-2s)(1-μ/{\bar\mu})^1/2,

C_N,s=(4-2s)/(N-2) ((N-2)/(N+2-2s)^{(N+2-2s)/(4-2s)}
and A_s=inf _{u∈ D^1,2(R^N)\{0}} {∫_R^N(|▽u|²-μu^2_/|x|²) dx}/{(∫_R^N|u|^2*(s)/|x|^s dx)^2/2*(s)}.

Key words: Critical Sobolev-Hardy exponent, elliptic equation, Mountain Pass theorem, subsuper solutions, nonhomogeneous