Existence,Multiplicity And Concentration Of Solutions For Choquard Equations

In this paper,we have two conclusion.First,we are interested in the existence,multiplicity and concentration behavior of nontrivial nonnegative solutions for the frac-tional Choqiard equation with critical exponent.Second,the Choquard equation with subcritical exponent has at least three solutions.We are interested in the existence,multiplicity and concentration behavior of non-trivial nonnegative solutions for the following fractional Choquard equation with critical exponent:?2s(-?)su+ V(x)u=??-N(1/|x|?*F(u)+|u|2s*-2u,x ? RN where ?>is a parameter,s ?(0,1),N>2s,2s*=2N/N-2s,0<?<min{2s,N-2s} and F(u)= ?0t f(?)d?.Assuming the global condition on V ? C(RN,R):(V0)0<V0:=inf x?RN V(x)<1im inf |x|?? V(x):=V?<+?,Concerning the function ? ? C(R,R),we assume that f(t)= 0 for t<0 and satisfies the following conditions:(f1)limt?0 f(t)/t=0.(f2)(?)q ?(2N-?/N,2N-?/N-2s)such that limt?? f(t)/tq-1=0(3)f(t)/t is increasing for every t>0.(f4)(?)??(qN,2N-?/N-2s),C>0 s.t.f(t)?ctù-1 for all t ?R+,where qN= max{2N-2s/N-2s,N+2s/N-2s}.Furthermore,we study the following Choquard equation by[35,Theorem 1.1]:where,?(?)R3 is an open,and bounded domain with a smooth boundary,h ? L2(?),0<?/<3,4<p<6,?>0,?>0.The nonlinearity ? ? C(R,R),??0,we assume that f(t)=0 for t<0 and satisfies the following conditions:(f1)limt?0 f(t)/t=0.(f2)There exists q ?(6-?/3,min{6-?,p/2})such that lim t??f(t)/tq-1=0.