For variational problems with prescribed mass, several methods commonly used to deal with unconstrained variational problems are no longer available. At the same time, a new critical exponent appeared, the L 2 -critical exponent. From the Gagliardo-Nirenberg inequality, we can find that the nonlinearity satisfying L 2 -subcritical or critical or supercritical leads to a different geometry for functional. In the case the functional is bounded below, i.e., L 2 subcritical case, the earliest result on this problem was obtained by Charles-Alexander Stuart using bifurcation methods to demonstrate the existence of solutions. Later, Thierry Cazenave and Pierre-Louis Lions obtained existence results using minimization methods. Then, Masataka Shibata considered Schrödinger equations with a general nonlinear term.