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〈μ^-=(N-2)^2/4, 0≤s〈2, where 2^*(s)=2(N-s)/N-2 is the critical Sobolev-Hardy expon...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〈μ^-=(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||*〈CN,sAs^N-s/4-2s(1-μ/μ)^1/2, CN,s=4-2s/N-2(N-2/N+2-2s)^N+2-2s/4-2s and As = inf u∈D^1,2(R^N)/{0}∫R^N(|△↓u|^2-μu^2/|x|^2)dx/(∫R^N|u|^2^*(s)/|x|^sdx)^2/2^*(s).展开更多
文摘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〈μ^-=(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||*〈CN,sAs^N-s/4-2s(1-μ/μ)^1/2, CN,s=4-2s/N-2(N-2/N+2-2s)^N+2-2s/4-2s and As = inf u∈D^1,2(R^N)/{0}∫R^N(|△↓u|^2-μu^2/|x|^2)dx/(∫R^N|u|^2^*(s)/|x|^sdx)^2/2^*(s).