Study On Normalized Solutions For Two Classes Of Nonlocal Problems

The main content of this master thesis includes four chapters:In Chapter 1,first of all,we introduce the background of two types of nonlocal problems:Kirchhoff equation and Fractional schrodinger equation and the current research situation.Then main results of this paper are introduced.In Chapter 2,we give a brief introduction to some of the notations,definitions and relative knowledge used in this paper.In Chapter 3,we study normalized solution of the following mass supercritical Kirchhoff equation with potential (?)where a,b>0,V is a potential function,14/3≤p<6,λ is the Lagrange multiplier.The existence of normalized solution of the above equation is discussed by mountains theorem and the lemma of global compactness if V satisfies some assumptions.Furthermore,the radial symmetry properties of solution are discussed.In Chapter 4,we investigate normalized solution of the following mass supercritical fractional Schrodinger equation with negative potential (?)where N≥ 2s,s∈(0,1),p ∈(2+4s/N,2s*),V>0 and(?).Firstly,for eachρ>0,we discuss the existence of a mountain pass solution the above equation at positive energy level and the nonexistence of solutions with negative energy,when V satisfy the corresponding assumptions.Secondly,when the mass p is smaller than some constant related to V,we prove the existence of a mountain pass solution of equation with positive energy.