Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities

The existence and nonexistence of the minimizer of the L²-constraint minimization problem e(α):=inf{E(u)|u∈H1(RN),∥u∥L2(RN)2=α} are studied. Here, E(u):=12∫RN|∇u|2+V(x)|u|2dx-∫RNF(|u|)dx,V(x) ∈ C(R^N) , 0 ≢ V(x) ≤ 0 , V(x) → 0 (| x| → ∞) and F(s)=∫0sf(t)dt is a rather general nonlinearity. We show that there exists α≥ 0 such that e(α) is attained for α> α and e(α) is not attained for 0 < α< α. We study differences between the cases V(x) ≢ 0 and V(x) ≡ 0 , and obtain sufficient conditions for α= 0. In particular, if N= 1 , 2 , then α= 0 , and hence e(α) is attained for all α> 0.