The Study Of The Existence Of Solutions For Two Fractional Kirchhoff Type Problem With Magnetic Field

In this paper,we discuss the existence of solutions for two fractional Kirchhoff type problem with magnetic field.Firstly,we consider the following magnetic field fractional p-Kirchhoff type problem with the critical Sobolev-Hardy and generalized Choquard nonlinearity (?) where Ω is a bounded smooth domain of RN containing 0,0≤α<ps<N with s ∈(0,1),A∈C(RN,RN)is a magnetic potential,M:R0+→R+ is a Kirchhoff function,f∈C1(R+,R),F(u)=∫0uf(t)dt.Here Iμ(x)=|x|-μ is the Riesz potential of orderμ∈(0,min{N 2ps},pα*=p(N-α)/N-ps is critical Sobolev-Hardy exponent.We establish the concentration compactness principle of p-fractional with magnetic field,and by using variational principle,we obtain the existence of nontrivial solutions in nondegenerate and degenerate cases.Secondly,we consider the magnetic field fractional(p,q)-Kirchhoff type problem with the critical Sobolev-Hardy (?) where Ω is a bounded smooth domain of RN containing 0,1<q<p,0≤α<ps<N with s ∈(0,1),A ∈ C(RN,RN)is a magnetic potential.pα*=p(N-α)/N-ps,whenα=0,p*=pN/N-ps is critical Sobolev exponent.1<r<pα*≤p*,k(x)∈Lp*/p*-r(Ω,C).We obtain the existence of nontrivial solutions by using variational principle.