The Existence Of Normalized Solutions For Several Classes Of Nonlocal Problems

There are many phenomenons related to nonlocal problems,for example,the irregular perturbations and the interaction of remote forces.Nonlocal problems attract more and more researchers.Integral-differential equations and equations with fractional Laplace operators are two typical nonlocal problems.In the study of partial differential equations,the existence of solutions is an important problem.This dissertation mainly considers the normalized solutions of nonlocal problems by variational methods.This paper is divided into seven chapters.In Chapter 1,the research background and research status are introduced.In Chapter 2,some preliminaries are given.In Chapter 3,the normalized solutions of the fractional Schrodinger equation with Sobolev critical exponent are considered.If N>2s,q∈(2,2+4s/N),c ∈(0,c0(μ)),then two solutions are obtained,one is the ground state solution obtained by the local minimizing methods,and the other is the mountain pass type solution obtained by the minimax principle.The L2strong convergence of the critical sequence is obtained by the sign of Lagrange multiplier,which is determined by Pohozaev identity together with I’μ(u)→0,this method is more concise than the methods used in[1],where in which the L2-strong convergence of the critical sequence is obtained by the subadditivity and continuity of m(c).If N>2s,q=2+4s/N or N≥(?),2+4s/N<q<2s*,then by Sobolev subcritical exponent approximating the Sobolev critical exponent,the mountain pass type solution is obtained.In Chapter 4,the normalized solutions of the fractional Schrodinger equation with Sobolev critical exponent and Choquard term are considered.If N≥ 2,α∈(N-2s,N),p∈[2,1+α+2s/N),c∈(0,c0(μ)),the ground state solution is obtained by the local minimizing methods;if 2s<N<6s,by the fine analysis of Choquard term,the estimates of the upper bound for the mountain pass level are obtained.Combined with the minimax principle,the existence of mountain pass type solution is obtained.In Chapter 5,the normalized solutions of fractional Schrodinger coupled equations with L2-supercritical nonlinearities are considered.If 2s<N ≤4s,1+2s/N<p<N/N-2s,then there is a constant β1>0 such that if 0<β<β1,then by two dimensional linking argument and the minimax principle,the existence of normalized solutions is obtained.There is a constant β2>0 such that if β>β2,then the existence of normalized solutions is obtained by mountain path structure,and the solution can be characterized as a minimal point of Rayleigh-type quotient.When the coupled equations combines the constraint conditions,there will be two Lagrange multipliers.Using Pohozaev identity and E’(u,u)→0,one of the signs for the two Lagrange multipliers is obtained.In order to determine the sign of another Lagrange multiplier,we prove a new result which is fractional Liouville type result.Based on this result and energy estimates,the sign of another Lagrange multiplier is determined,and then the compactness of the critical sequence is obtained.In Chapter 6,the normalized solutions of fractional Schrodinger coupled equations with Sobolev critical exponent are considered.When N>2s、2<q<2 +4s/N,if s∈(0,1).and there exists a constant Ω0 such that Ω(c,d)∈(0,Ω0),then the ground state solution is obtained by Ekeland variational principle;if s ∈(1/2,1),and there exists a constant Ω0 such that Ω(c,d)∈(0,Ω0),then the mountain path type solution is obtained by the minimax principle.When s ∈(0,1),2s<N≤4s,q=2+4s/N or(?)≤N ≤4s,2+4s/N<q<2s*,by the minimax principle,the mountain path type solution is obtained.In this chapter,the compactness of the critical sequence is proved by the weak monotonicity of m(c,d)and the Liouville type results given in Chapter 5.When proving the existence of mountain path type solutions for q ∈(2,2+4s/N).the estimates of the upper bound for M(c,d)are proved by constructing two paths(uc+tUε,vd)and(uc,vd+tUε),combining with the decay estimates of the gradient for solution and using Pohozaev identity.In Chapter 7,the research results of this thesis are summarized and the further research plans are discussed.