Normalized solution to the Schrödinger equation with potential and general nonlinear term: Mass super-critical case

By Yanheng Ding, X. Zhong. Journal of Differential Equations, 2021

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In present paper, we prove the existence of solutions (λ, u) ∈ R×H1(RN ) to the following Schrödinger equation { −∆u(x) + V (x)u(x) + λu(x) = g(u(x)) in R 0 ≤ u(x) ∈ H(R ), N ≥ 3 satisfying the normalization constraint ∫ R udx = a. We treat the so-called mass super-critical case here. Under an explicit smallness assumption on V and some Ambrosetti-Rabinowitz type conditions on g, we can prove the existence of ground state normalized solutions for prescribed mass a > 0. Furthermore, we emphasize that the mountain pass characterization of a minimizing solution of the problem inf {∫ [ 1 2 |∇u| + 1 2 V (x)u −G(u) ] dx : ‖u‖L2(RN ) = a, P [u] = 0 } , where G(s) = ∫ s 0 g(τ)dτ and P [u] = ∫ [ |∇u| − 1 2 〈∇V (x), x〉u −N ( 1 2 g(u)u−G(u) )] dx. 2010 Mathematics Subject Classification: 35Q55,35Q51, 35B09,35B32,35B40.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ba430e44223e1069a440d4c20ed940f96ce12621