2021/04/26 by Riccardo Molle, Molle, Riccardo, Giuseppe Riey +3 · 3 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2104.12834
openalex publication_date 2021/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the existence of positive solutions with prescribed L2-norm for the Schrödinger equation -Δu-V(x)u+λu=|u|p-2u λ∈ ℝ, u∈ H1(ℝN), where V≥ 0, N≥ 1 and p∈(2+\frac 4 N,2^*), 2^*:=(2N)/(N-2) if N≥ 3 and 2^*:=+∞ if N=1,2. We treat two cases. Firstly, under an explicit smallness assumption on V and no condition on the mass, we prove the existence of a mountain pass solution at positive energy level, and we exclude the existence of solutions with negative energy. Secondly, requiring that the mass is smaller than some explicit bound, depending on V, and that V is not too small in a suitable sense, we find two solutions: a local minimizer with negative energy, and a mountain pass solution with positive energy. Moreover, a nonexistence result is proved.