Bound states for logarithmic Schrodinger equations with potentials unbounded below

2019/05/16 by Chengxiang Zhang, Xu Zhang, Zhang, Chengxiang +1
Computer Science · Mathematics · #35B25 #35Q55 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1905.06687

openalex publication_date 2019/05/16 · openalex created_date 2019/05/29 · openalex updated_date 2026/07/28

Abstract

We study the existence and concentration behavior of the bound states for the following logarithmic Schrödinger equation \begincases -ε²Δv+V(x)v=vlog v² amp;\text in \mathbb R^N,
v(x)→ 0 amp;\text as |x|→∞, \endcases where N≥ 1, ε>0 is a small parameter, and V may be unbounded below at infinity with a speed of at most quadratic strength. We show that around various types of local topological critical points of the potential function, positive bound state solutions exist and concentrate as ε→0.

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