Bulletin of the
Korean Mathematical Society BKMS

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).