Normalized solutions for the NLS equation with potential in higher dimension: the purely Sobolev critical case

2025/07/31 by Sun, Juntao, Yao, Shuai, Zhang, He
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.23639

Abstract

We study normalized solutions for the nonlinear Schrodinger (NLS) equation with potential and Sobolev critical nonlinearity. By establishing suitable assumptions on the potential, together with new techniques, we find a mountain-pass type solution for N>=6, which solves an open problem presented in a recent paper [Verzini and Yu, arXiv:2505.05357v1]. Moreover, we also find a local minimizer with negative energy for N>=3, which improves the results in [Verzini and Yu, arXiv:2505.05357v1].

Citations

• Normalized solutions for the nonlinear Schrödinger equation with potential: the purely Sobolev critical case
• Normalized solutions for Sobolev critical Schrödinger equations on bounded domains
• Normalized solution to the Schödinger equation with potential and general nonlinear term: Mass super-critical case
• Normalized solutions to mass supercritical Schrodinger equations with negative potential
• Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities
• Multiple normalized solutions for a Sobolev critical Schrödinger equation
• Normalized concentrating solutions to nonlinear elliptic problems
• Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case
• Normalized ground states for the NLS equation with combined nonlinearities
• Normalized bound states for the nonlinear Schrodinger equation in bounded domains

Related

• Normalized solutions for the nonlinear Schrödinger equation with potential: the purely Sobolev critical case
• Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case
• Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities
• Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity
• Normalized solutions for the Sobolev critical Schrödinger equation with trapping potential