Normalized Solutions For Two Classes Of Nonlinear Choquard Equations

In this paper,we study the existence of normalized solutions for two classes of non-linear Choquard equations by using variational methods.Firstly,we consider the existence of normalized solutions for the Choquard equation with combined nonlinearities.Secondly,we discuss the existence of normalized ground states for the fractional Choquard equation with L2-critical exponent.The main theoretical bases are the methods of the minimizing se-quences,the vanishing lemma,the minimax principle,Hardy-Littlewood-Sobolev inequality,the Pohozaev identity and some analytical skills.Firstly,we discuss the existence of normalized solutions for the following nonlinear Choquard equation with combined nonlinearities（?） where N≥3,（?）,λ∈R is lagrange multiplier,μ∈R.The main results are as follows:Theorem 3.1.1.Let N≥3,（?）,c,μ>0,and（?）where γ_t=（Nt-N-α）/t,t=p,q.Then the problem（P1）has weak solutions （?） with （?）.Theorem 3.1.2.Let N≥3,α∈（0,N）,（?）,c>0,μ<0,and（?）where γp,γq are given in Theorem 3.1.1.Then the problem（P1）has weak solutions （?） with （?）.Secondly,we consider the existence of normalized ground states for L2-critical exponent p=（N+α+2s）/N of the following fractional Choquard equation（?） where N≥3,s∈E（0,1）,α∈（0,N）,κα（x）=|x|^α-N,λ∈R is lagrange multiplier.We can prove that for p∈（（N+α）/N,（N+α）/（N-2s））,（?）where equality holds for u=φ_p,where φ_p is the ground state of the equation（?）Moreover,when p=（N+α+2s）/N,all groundstate solutions of the equation（?） have the same L2 norm（see Lemma 4.1.1 and Lemma 4.1.3）.Set （?）.The main result is as follows:Theorem 4.1.1.Let N≥3,s∈（0,1）,α∈（N-2s,N）and p=（N+α+2s）/N.（1）If 0<c<c*,then problem（P2）has no normalized ground state;（2）If c=c*,then each groundstate of the equation（*）is a normalized ground state of the problem（P2）;（3）If c>c*,then problem（P2）has no normalized ground state.