Normalized solutions of mass supercritical Schrödinger–Poisson equation with potential - Peng, Xueqin; Rizzi, Matteo (Calculus of variations and partial differential equations, 2025)
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.