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:ε2s(-△)su+ V(x)u=εμ-N(1/|x|μ*F(u)+|u|2s*-2u,x ∈ RN where ε>is a parameter,s ∈(0,1),N>2s,2s*=2N/N-2s,0<μ<min{2s,N-2s} and F(u)= ∫0t f(τ)dτ.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-μ/N,2N-μ/N-2s)such that limt→∞ f(t)/tq-1=0(3)f(t)/t is increasing for every t>0.(f4)(?)σ∈(qN,2N-μ/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,Ω(?)R3 is an open,and bounded domain with a smooth boundary,h ∈ L2(Ω),0<μ/<3,4<p<6,β>0,λ>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-μ/3,min{6-μ,p/2})such that lim t→∞f(t)/tq-1=0.