Normalized clustering peak solutions for Schrödinger equations with general nonlinearities

• Published in: Calculus of variations and partial differential equations 2024-12, Vol.63 (9), Article 220
• Main Authors: Zhang, Chengxiang, Zhang, Xu
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Centroids; Clustering; Concavity; Control; Gradient flow; Ground state; Lagrange multiplier; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Nonlinearity; Schrodinger equation; Systems Theory; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-024-02830-5

We are concerned with the normalized ℓ -peak solutions to the nonlinear Schrödinger equation - ε 2 Δ v + V ( x ) v = f ( v ) + λ v , ∫ R N v 2 = α ε N . Here λ ∈ R will arise as a Lagrange multiplier, V has a local maximum point, and f is a general L 2 -subcritical nonlinearity that satisfies a nonlipschitzian property such that lim s → 0 f ( s ) / s = - ∞ . The peaks of solutions that we construct cluster around a local maximum of V as ε → 0 . Since there is no information about the uniqueness or nondegeneracy of the limiting system, a sensitive lower gradient estimate should be made when the local centroids of the functions are away from the local maximum of V . We introduce a new method to obtain this estimate, which differs significantly from the ideas of del Pino and Felmer [ 22 ] (Math. Ann. 2002), where a special gradient flow with high regularity is used, and in Byeon and Tanaka [ 7 , 8 ] (J. Eur. Math. Soc. 2013 & Mem. Amer. Math. Soc. 2014), where an additional translation flow is introduced. We also give the existence of ground state solutions for the autonomous problem, i.e., the case V ≡ 0 . The ground state energy is not always negative and the strict subadditivity of the ground state energy is achieved here by strict concavity.