Normalized solutions for the Sobolev critical Schrödinger equation with trapping potential

2025/11/24 by Yu, Junwei
Mathematics · #35B33 #35J20 #35J61 #35Q55 #35Q89 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · doi:10.48550/arxiv.2511.19271

openalex publication_date 2025/11/24 · openalex created_date 2025/11/27 · openalex updated_date 2026/07/28

Abstract

We study the existence and multiplicity of positive normalized solutions with prescribed L²-norm for the Sobolev critical Schrödinger equation -ΔU + V(x) U = λU + |U|^{2^*-2} U in ℝ^N, ∫_ℝ^N U² dx = ρ², where N ≥ 3, V≥ 0 is a trapping potential, λ∈ ℝ and 2^*=(2N)/(N-2). Our first result is that the existence of local minimum solutions for ρ∈ (0, ρ^*), for some suitable ρ^* > 0, under appropriate assumptions on the potential. These solutions correspond to ground states. Our second result concerns the existence of mountain pass solutions, under the same assumptions.

Citations

• Energy local minimizers for the nonlinear Schrödinger equation on product spaces
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• Normalized bound states for the nonlinear Schrodinger equation in\n bounded domains
• Stable solitary waves with prescribed L²-mass for the cubic\n Schr "odinger system with trapping potentials
• Nonlinear scalar field equations, I existence of a ground state
• Orbital stability of standing waves for some nonlinear Schr�dinger equations

Related

• Normalized solutions for Sobolev critical Schrödinger equations on bounded domains
• Normalized solutions for nonlinear Schrödinger equation involving potential and Sobolev critical exponent
• Normalized solutions for a Sobolev critical quasilinear Schrödinger equation
• Normalized solutions for the nonlinear Schrödinger equation with potential: the purely Sobolev critical case
• Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities