Normalized solutions of mass supercritical Schrödinger–Poisson equation with potential

• Published in: Calculus of variations and partial differential equations Vol. 64; no. 5
• Main Authors: Peng, Xueqin, Rizzi, Matteo
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2025; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Lagrange multiplier; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Poisson equation; Systems Theory; Theoretical; 35J10; 35J50; 35J60
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-025-03012-7

In this paper we prove the existence of normalized solutions ( λ , u ) ⊂ ( 0 , ∞ ) × H 1 ( R 3 ) to the following Schrödinger–Poisson equation - Δ u + V ( x ) u + λ u + ( | x | - 1 ∗ u 2 ) u = | u | p - 2 u in R 3 , u > 0 , ∫ R 3 u 2 d x = a 2 , where a > 0 is fixed, p ∈ ( 10 3 , 6 ) is a given exponent and the potential V satisfies some suitable conditions. Since the L 2 ( R 3 ) -norm of u is fixed, λ appears as a Lagrange multiplier. For V ( x ) ≥ 0 , our solutions are obtained by using a mountain-pass argument on bounded domains and a limit process introduced by Bartsch et al (Commun Partial Differ Equ 46:1729–1756, 2021). For V ( x ) ≤ 0 , we directly construct an entire mountain-pass solution with positive energy.