SOLUTIONS FOR A NONHOMOGENEOUS ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENT IN RN*

For the following elliptic problem {-Δu-μu|x|2=|u|2*(s)-2u|x|s+h(x),  on RNu∈D1,2(RN),N≥3, 0≤μ<μ¯=(N-2)24, 0≤s<2, where 2*(s)=2(N-s)N-2 is the critical Sobolev-Hardy exponent, h(x)∈(D1,2(RN)*h(x)∈(D1,2(RN)*, the dual space of (D1,2(RN))(D1,2(RN)), 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,sAsN-s4-2s(1-μμ)12,CN,s=4-2sN-2(N-2N+2-2s)N+2-2s4-2s and As=infu∈D1,2(RN)\{0}∫RN(|∇u|2-μu2|x|2)dx(∫RN|u|2*(s)|x|sdx)22*(s).