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<?<N and 2?*is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.V(x)is a continuous function such that the spectrum?(-? + 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(?)?(-?+V)and ?(-?+V)?(-?,0)?(?)where ? 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|?).(Gi):g? C(RN×R,R)and |g(x,s)|??(x)|s|+h(x)|s|p-1,where ?(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.