Another look at Schrödinger equations with prescribed mass

Abstract

In this paper, we investigate the existence of solutions for the nonlinear Schrödinger equation −Δu−λu=f(u) in RN with an L2-constraint in the Sobolev subcritical case when f possesses several weaker L2-supercritical conditions and in the Sobolev critical case when f(u)=μ|u|q−2u+|u|2⁎−2u with μ>0 and 2<q<2⁎=2NN−2 allowing to be L2-subcritical, L2-critical or L2-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 L2-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.