Existence And Properties Of Solutions For Choquard Type Elliptic Equations

In this paper7 we mainly study the existences and properties of solutions to equations involving Choquard type in RN.The main work consists of four parts,as follows.Firstly,we consider the existence of the ground state to some Choquard type equation using scaling,transformation and some special minimization method.We extend some kind of interpolation inequality involving Gagliardo-Nirenberg type and determine the best variational constant C(p,q,?,N).The interpolation inequality involving Gagliardo-Nirenberg type,as follows:(?)where 0<?<N,q ?(pN-?/N,P/22N-?/N-p)for 2?p<N;q?(pN-?/N,+?)for p=N.And A=2N(q-p)+p?/p2,B=2qp-[2N(q-p)+p?]/p2.Secondly,we introduce a kind of Sobolev compact theorems involving weaker potential functions.Along with Nehari splitting method,we prove the existences of two positive solutions for the following Choquard type equation,(?)where 1<q<2?p<2r,0<?<N,??R+\{0}.r?(pN-?/N,P2N-?/2 N-p))for p<N;r ?(pN-?/N,+?)for p= N.h(x)and ?(x)can change sign,V(x)satisfies some certain conditions.Thirdly,we consider the existence of the non-radially symmetric ground state for p-Laplacian equations involving Choquard type.-?pu+|u|p-2u=(Ia*|u|r)|u|r-2u+|u|q-2u,x?RNwhere N?3,??(0,N),p?2,p/2 N+?/N<r<p/2 N+?/N-p,p<q<p*.And for any x ? RN\{0},(?).Determining the regular information of the nonlinear parts for the equation,we get the existence of positive solutions for the equation by means of the Pohozaev manifold constraints,minimax theory and deformation lemma.Along with the technique developed by Jeanjean and Tanaka,we prove that the positive solution is a ground state.When p=2 and N=3.by a variant variational identity and a constraint set,we can prove the existence of a non-radially symmetric solution.Moreover,this solution u(x1,x2,x3)is radially symmetric with respect to(x1,x2)and odd with respect to x3.Finally,by means of the Pohozaev manifold constraint and minimax theory,we prove that the existence of ground state for p-Laplacian equation involving Choquard type which is different from the third part.-?pu+|u|p-2u=(Ia*|u|r)|u|r-2u+(I?*|u|q)|u|q-2u,x?RN where N?3,? ?(0,N),??(0,N),p ?2,p/2 N+?/N<r<p/2 N+?/N-p,p/2 N+?/N<q<p/2 N+?/N-p.And(?),for any(?).