Some results on the constrained variational problem

This survey talk mainly involves two parts: The normalized solution and the stability of inequality. The first part will focus on the existence of solutions to some Schrodinger systems or equation satisfying the normalization constraint, including the normalized solutions to the BEC-type systems with a wider range of parameters and the existence of normalized solutions of nonlinear Schrodinger equations with combined nonlinearities in large bounded domains. The second part of the talk is concerned with the quantitative stability for several inequality, such as Sobolev trace inequality and CKN-inequality.