L~2-Normalized Least Energy Solution For Chern-Simons-Schr(?)dinger Equation

In this thesis,we mainly study the existence of solutions with given L² norm for a class of Chern-Simons-Schrodinger（CSS）equations in R².This kind of problem can be transformed into a minimization problem of the corresponding energy functional E_p^β（u）under the constraint of ‖u‖_L²（R²）=1.In this thesis,we are concerned with the existence of minimizers and their properties for the constrained minimization problem.Our main results are as follows:In Chapter 2,we study the constrained variational minimization problem under V（x）≡0.To prove this minimization problem is attained,the main difficulty is the lack of compactness for the embedding from H_r¹（R²） to L²（R²）.To overcome this difficulty,we use the Ekeland’s variational principle and the fact that the minimum of the minimization problem is negative.For the mass subcritical case,that is,p∈（0,2）,we prove that the constrained variational minimization can be attained;For the mass-critical case,that is p=2,we find two positive constantsβ^*>β_*,which can be given explicitly,such that the constrained variational minimization problem with V（x）≡0 can’t be attained for both β>β^*and β≤β_*.In Chapter 3,we study the constrained variational minimization problem under V（x）（?）0 being coercive.In this case,the embedding from to H_r¹（R²）to L²（R²）is compact.In this case,the main difficulty in getting the minimizers for the minimization problem is to verify the boundedness of minimizing sequence.By contradiction,we prove that the constrained variational minimization problem can be attained if p∈（0,2）,but for p=2,The minimization problem can be attained forβ≤β_*and can’t be attained for β>β^*.In Chapter 4,under V（x）≥0 and the mass-subcritical case（i.e.,p ∈（0,2））,we discuss the limit behavior of the constrained minimum energy at p→2.