Normalized solutions for nonlinear Schrödinger equation involving potential and Sobolev critical exponent

2023/12/26 by Zhen‐Feng Jin, Jin, Zhen-Feng, Weimin Zhang +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2312.15917

openalex publication_date 2023/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the existence of positive solutions with prescribed L²-norm for the following nonlinear Schrödinger equation involving potential and Sobolev critical exponent \begincases -Δu+V(x)u=λu+μ|u|^p-2u+|u|^(4)/(N-2)u \text in ℝ^N,
‖u‖₂=agt;0,
\endcases where N≥ 3, μ>0, p∈ [2+(4)/(N), (2N)/(N-2)) and V∈ C¹(ℝ^N). Under different assumptions on V, we derive two different Pohozaev identities. Based on these two cases, we respectively obtain the existence of positive solution. As far as we are aware, we did not find any works on normalized solutions with Sobolev critical growth and potential V \not≡ 0. Our results extend some results of Wei and Wu [J. Funct. Anal. 283(2022)] to the potential case.

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