On The Existence Of Solutions To Three Classes Of Fractional Schr(?)dinger-Poisson Systems

In this paper,we study the following three kinds of fractional Schr(?)dinger-Poisson systems with different potential:by making appropriate assumptions on the parameters in the system,we obtain the existence of solutions of different potentials.We concern the first class of fractional Schr(?)dinger-Poisson system with Hardy potential.Where s ∈(3/4,1),V(x)=w-β/|x|2s,w ∈ R,λ,β>0 are parameters,f(u)=|u|p-1u,p ∈(7/3,3+2s/3-2s).We obtain the existence of the normalized solution by the constrained variational methods.We consider the second class of fractional Schrodinger-Poisson system with coercive potential.Where s ∈(3/4,1),V ∈ C(RN,R0+)(where R0+=[0,+∞))is a coercive potential(i.e.,(?)),λ>0 is a parameter,f(u)=|u|p-1u,p ∈(1,2).We will prove that there exists λ0>0 such that the above system has at least two positive solutions uλ0 and uλ1 for any λ ∈(0,λ0).Furthermore,uλ0 is a ground state(i.e.,the least energy solution)which must blow up as λ→0.We consider the third class of fractional Schr(?)dinger-Poisson system with coercive sighchanging potential.In this case,let λ≡1.Where s ∈(3/4,1),V is a coercive sign-changing potential so that the fractional Schr(?)dinger operator(-Δ)s+V is indefinite,f(u)is a general nonlinearity.According to the local linking theorem,we find that the system has at least two non-trivial solutions when 3/4<s<1.