Existence Of Global Solutions For Indefinite Linear Choquard Equations With Hardy-Littlewood-Sobolev Critical Exponents

The main goal of this thesis is to consider existence of global solusions for a class of indefinite nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents,we apply variational methods to prove the global solusions for this problem.where N≥3,0<μ<N and 2μ*is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.V(x)is a continuous function such that the spectrumσ(-△ + V(x))of-△+ V(x)in L2(RN)has a negative part,K(x)is a bounded positive function,g is of subcritical growth,and satisfying the following conditions:(V1):V(x)∈C(RN)∩L∞(RN)and liminf|x|→∞V(x)=v∞>0.(V2):(W1(x)-v∞)∈LN/2(RN),0(?)σ(-△+V)and σ(-△+V)∩(-∞,0)≠(?)where σ denotes the spectrum in L2(RN)and W1(x)=max{V(x),v∞}.(K1):K(x)∈ C(RN)attains its maximum at 0.KM:K(0)=maxRN K(x)and there exist positive constants Kmin and a such that K(x)≥Kmin and K(0)-K(x)=O(|x|α).(Gi):g∈ C(RN×R,R)and |g(x,s)|≤ω(x)|s|+h(x)|s|p-1,where ω(x)∈ LN/2(RN)∩L∞(RN),2<p<2*and h(x)∈L2*/2*-P(RN)∩L∞(RN).(G2):lims→0 g(x,s)/s = 0 uniformly on RN.(G3):0≤2G(x,s)≤sg(x,s)for a.e.x∈RN,(?)s∈R,where G(x,s):=∫0s g(x,t)dt.