Normalized Solutions for Schrödinger Equations with Stein–Weiss Potential of Critical Exponential Growth

• Published in: The Journal of geometric analysis Vol. 33; no. 10
• Main Authors: Yuan, Shuai, Tang, Xianhua, Chen, Sitong
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2023; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Critical point; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Mathematical analysis; Mathematics; Mathematics and Statistics; Minimax technique; Nonlinear equations; Schrodinger equation; Variational methods; 35J20; Critical exponential growth; 35J62; 35Q55; Nonlinear Schrödinger equation; Trudinger–Moser inequality; Stein–Weiss potential
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-023-01396-6

In this paper, we focus on the existence of normalized solutions to the following Schrödinger equation with the Stein–Weiss potential - Δ u + λ u = I μ × F ( u ) | x | α f ( u ) | x | α , x ∈ R 2 , where 2 α + μ ≤ 2 , 0 < μ < 2 , I μ denotes the Riesz potential and f : R → R has critical exponential growth which behaves like e α u 2 . The solutions correspond to critical points of the underlying energy functional subject to the L 2 -norm constraint, namely, ∫ R 2 | u | 2 d x = a 2 for a > 0 given. Under some weak assumptions, we prove the existence of the normalized solution for the equation by developing refined variational methods. In particular, we shall establish two new approaches to estimate precisely the minimax level of the underlying energy functional. As far as we know, our result is the first one in seeking normalized solutions of nonlinear equations involving the nonlocal Stein–Weiss reaction.