Infinitely many solutions to elliptic systems involving critical exponents and Hardy potential

摘要

In this paper, we consider the following elliptic systems involving critical Sobolev growth and Hardy potential: -Δu-λ1u|x|2= a1|u|2~*-2u+bh(x)αα+β|u|α-2u|v|β,x∈RN, -Δv-λ2v|x|2=a2|v|2~*-2v+bh(x)βα+β|u|α |v|β-2v,x∈RN, where N ≥ 3,λ_1,λ_2 ∈ [0,ΛN), ΛN:=N-222 is the best Hardy constant. 2~*=2NN-2 is the critical Sobolev exponent. a_1,a_2, b are positive parameters, α,β > 0 and 1 < α + β: = q < 2 < 2~*. h(x)∈Lq′(RN) with q′=2~*2~*-q. By means of the concentration-compactness principle and Kajikiya's new version of symmetric mountain pass lemma, we obtain infinitely many solutions that tend to zero for suitable positive parameters a_1,a_2,b and λ_1,λ_2.