2024/09/08 by Shubin Yu, Chen Yang, Yu, Shubin +3
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2409.05130
openalex publication_date 2024/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation \-(a+b∫Ω|∇ u|2dx)Δu+V(x)u=μu+β^*|u|(8)/(3)u · amp;in Ω,
u=0 · amp;on ∂Ω,
∫Ω|u|2dx=1, . where a≥0, b>0, the function V(x) is a trapping potential in a bounded domain Ω⊂\mathbb R3, β^*:=(b)/(2)|Q|2(8)/(3) and Q is the unique positive radially symmetric solution of equation -2Δu+(1)/(3)u-|u|(8)/(3)u=0. We consider the existence of constraint minimizers for the associated energy functional involving the parameter a. The minimizer corresponds to the normalized ground state of above problem, and it exists if and only if a>0. Moreover, when V(x) attains its flattest global minimum at an inner point or only at the boundary of Ω, we analyze the fine limit profiles of the minimizers as a\searrow 0, including mass concentration at an inner point or near the boundary of Ω. In particular, we further establish the local uniqueness of the minimizer if it is concentrated at a unique inner point.