Another look at Schrödinger equations with prescribed mass.

In this paper, we investigate the existence of solutions for the nonlinear Schrödinger equation − Δ u − λ u = f (u) in R N with an L 2 -constraint in the Sobolev subcritical case when f possesses several weaker L 2 -supercritical conditions and in the Sobolev critical case when f (u) = μ | u | q − 2 u + | u | 2 ⁎ − 2 u with μ > 0 and 2 < q < 2 ⁎ = 2 N N − 2 allowing to be L 2 -subcritical, L 2 -critical or L 2 -supercritical, where N ≥ 3 is the dimension, λ ∈ R and f ∈ C (R , R). By establishing several new critical point theorems on a manifold, we introduce a new variational approach which enables us to weaken the previous L 2 -supercritical conditions in the Sobolev subcritical case and also to present an alternative scheme for all 2 < q < 2 ⁎ , which is technically simpler compared to the Ghoussoub minimax principle [7] involving topological arguments, to construct bounded (PS) sequences on a manifold when the reaction performs as a mixed dispersion. In particular, the analysis developed in this paper also allows to control the energy level in the Sobolev critical case in a unified way whether N = 3 or N ≥ 4. Furthermore, this new approach can provide a new look at what were expected by Soave [13] and by Jeanjean-Le [10] and may it can be applied and modified to more related variational problems. [ABSTRACT FROM AUTHOR]