Concentration Behavior of Ground States for L²-Critical Schrödinger Equation with a Spatially Decaying Nonlinearity

2021/06/28 by Yong Luo, Shu Zhang, Luo, Yong +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.2106.14375

openalex publication_date 2021/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the following time-independent nonlinear L²-critical Schrödinger equation -Δu(x)+V(x)u(x)-a|x|^-b|u|^1+(4-2b)/(N)=μu(x) \hboxin ℝ^N, where μ∈ℝ, a&gt;0, N≥ 1, 0<b>0 such that minimizer exists for 0a^*. However if a=a^*, it is proved that whether minimizer exists depends sensitively on the value of V(0). Moreover, when there is no minimizer at threshold a^*, we give a detailed concentration behavior of minimizers as a\nearrow a^*, based on which we finally prove that there is a unique minimizer as a\nearrow a^*.</b>

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