Existence,multiplicity And Concentrate Behavior Of Solutions For Magnetic Equations

In this thesis,We focus our attention on equations with magnetic field.First,we study the existence and concentrate behavior of solutions for following non-linear fractional Schrodinger equation with magnetic field (?) where ε>0 is a small parameter,0<s<1,2<p<2*=6/3-2s,(-△)As is the fractional magnetic Laplacian,V,K are both positive potentials.A(x)is a magnetic potential.When V,K,A satisfy suitable assumpt.ions,we prove the existence of a nontrivial solution by using variational methods for each ε>0 sufficiently small,and we complete the proof of concentrate behavior of solutions.Next,we use variational methods,combine with saddle point theory to study the following problem:(?)where Ω(?)R3 is a bounded open set with smooth boundary,A =(A1,A2,……An):is a magnetic field,such that A E Lloc2(RN),▽A:=-i▽ + A,-△A:=(-i▽+A)2.V(x)≥0 and it is continuous.And we implied that there are three solutions in this problem when f,V,h satisfy suitable assumptions.