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

Abstract

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