Bull. Korean Math. Soc. 2024; 61(2): 541-556
Online first article March 7, 2024 Printed March 31, 2024
https://doi.org/10.4134/BKMS.b230226
Copyright © The Korean Mathematical Society.
Yongkuan Cheng, Yaotian Shen
South China University of Technology; South China University of Technology
In this paper, we consider a model problem arising from a classical planar Heisenberg ferromagnetic spin chain \begin{equation*} -\Delta u+V(x)u-\frac{u}{\sqrt{1-u^2}}\Delta \sqrt{1-u^2}=\lambda |u|^{p-2}u,\ x\in\mathbb{R}^{N}, \end{equation*} where $2\leq p<2^*, N\geq 3$. By the Ekeland variational principle, the cut off technique, the change of variables and the $L^{\infty}$ estimate, we study the existence of positive solutions. Here, we construct the $L^{\infty}$ estimate of the solution in an entirely different way. Particularly, all the constants in the expression of this estimate are so well known.
Keywords: Schr\"{o}dinger equations, $L^{\infty}$ estimate, Heisenberg ferromagnet
MSC numbers: Primary 35B33, 35J20, 35J60, 35Q55
Supported by: This work was financially supported by NSFC (No.~12271179), the Guangdong Basic and Applied Basic Research Foundation (No.~2020A1515010338).
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