Quasilinear Schrödinger equations with superlinear terms describing the Heisenberg ferromagnetic spin chain

In this paper, we consider a model problem arising from a classical planar Heisenberg ferromagnetic spin chain: $-\mathrm{\Delta }u+V(x)u-\frac{u}{\sqrt{1-u^2}}\mathrm{\Delta }\sqrt{1-u^2}=c|u{|}^{p-2}u,x\in {\mathbb{R}}^N,$ \( -\Delta u+V(x)u-\frac{u}{\sqrt{1-u^{2}}}\Delta \sqrt{1-u^{2}}=c \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}, \) where $2<p<2^{\ast }$ $2< p<2^{*}$ , $c>0$ $c>0$ and $N\ge 3$ $N\geq 3$ . By the cutoff technique, the change of variables and the $L^{\mathrm{\infty }}$ $L^{\infty}$ estimate, we prove that there exists $c_0>0$ $c_{0}>0$ , such that for any $c>c_0$ $c>c_{0}$ this problem admits a positive solution. Here, in contrast to the Morse iteration method, we construct the $L^{\mathrm{\infty }}$ $L^{\infty}$ estimate of the solution. In particular, we give the specific expression of $c_0$ $c_{0}$ .