A new observation for the normalized solution of the Schrödinger equation

We consider the following nonlinear Schrödinger equation in R N ( N ≥ 2 ) : - Δ u + λ u = g ( u ) , u ∈ H 1 ( R N ) , ∫ R N u 2 = c , where c > 0 is a given constant, λ ∈ R is a Lagrange multiplier, and g ∈ C 1 ( R , R ) . We deal with the case where the associated functional is not bounded from below on the L 2 sphere S ( c ) = u ∈ H 1 R N : ∫ R N u 2 = c . We show that the ground state energy is strictly decreasing with respect to c . Then we apply this property to give a new proof for the existence of ground state solutions via minimizing methods. We also obtain some other properties of the ground state energy.