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

Abstract

Abstract In this paper, we consider a model problem arising from a classical planar Heisenberg ferromagnetic spin chain: − Δ u + V ( x ) u − u 1 − u 2 Δ 1 − u 2 = c | u | p − 2 u , x ∈ 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 0 $c>0$ and N ≥ 3 $N\geq 3$ . By the cutoff technique, the change of variables and the L ∞ $L^{\infty}$ 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 ∞ $L^{\infty}$ estimate of the solution. In particular, we give the specific expression of c 0 $c_{0}$ .

Keywords

• Schrödinger equations
• L ∞ $L^{\infty}$ estimate
• Heisenberg ferromagnet