Existence Of Multiple Positive Solutions For Fractional Laplace Problems With Critical Growth And Sign-changing Weight

We consider the following systems:(?)where 1<q<2,2s*=2N/N-2s(N>2s),s ∈(0,1),q*=2s*/2s*-q,Ω(?)RN is a bound-ed domain with smooth boundary.The decomposition of Nehari manifolds is studied with concave-convex nonlinearity.Furthermore,by using this result and Lusternik-Schnirelman category,we prove that there are at least three positive solutions for(Ef)if(D1)-(D3)satisfy:(Dl)there exist δ0>0,such that BN(2δ0)\(?),where BN(r)={x∈RN||x|<r};(D2)there exist δ>0 and δ<δ0 such that BN(δ)∩ Ω =(?);(D3)f(x)=f+(x)-f-(x),where f±:Ω→R is a continuous function,and there is a continuous differentiable domain Ω of first order,BN(2δ0)\(?)satisfy:(a)(?)x∈(?),f-(x)=0,f+(x)>0(b)(?)x∈Ω\(?),f-(x)≥0,f+(x)=0.