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

IF 1.7 4区 数学 Q1 Mathematics
Yongkuan Cheng, Yaotian Shen
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引用次数: 0

Abstract

In this paper, we consider a model problem arising from a classical planar Heisenberg ferromagnetic spin chain: $$ -\Delta u+V(x)u-\frac{u}{\sqrt{1-u^{2}}}\Delta \sqrt{1-u^{2}}=c \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}, $$ where $2< p<2^{*}$ , $c>0$ and $N\geq 3$ . By the cutoff technique, the change of variables and the $L^{\infty}$ estimate, we prove that there exists $c_{0}>0$ , such that for any $c>c_{0}$ this problem admits a positive solution. Here, in contrast to the Morse iteration method, we construct the $L^{\infty}$ estimate of the solution. In particular, we give the specific expression of $c_{0}$ .
描述海森堡铁磁自旋链的带超线性项的准线性薛定谔方程
在本文中,我们考虑了一个由经典平面海森堡铁磁自旋链产生的模型问题:$$ -\Delta u+V(x)u-\frac{u}{\sqrt{1-u^{2}}}\Delta \sqrt{1-u^{2}}=c \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}, $$其中$2< p0$,$N\geq 3$。通过截断技术、变量变化和 $L^{infty}$ 估计,我们证明存在 $c_{0}>0$ ,这样对于任意 $c>c_{0}$ 问题都有一个正解。在这里,与莫尔斯迭代法不同,我们构建了解的 $L^{\infty}$ 估计值。我们特别给出了 $c_{0}$ 的具体表达式。
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来源期刊
Boundary Value Problems
Boundary Value Problems MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.00
自引率
5.90%
发文量
83
审稿时长
4 months
期刊介绍: The main aim of Boundary Value Problems is to provide a forum to promote, encourage, and bring together various disciplines which use the theory, methods, and applications of boundary value problems. Boundary Value Problems will publish very high quality research articles on boundary value problems for ordinary, functional, difference, elliptic, parabolic, and hyperbolic differential equations. Articles on singular, free, and ill-posed boundary value problems, and other areas of abstract and concrete analysis are welcome. In addition to regular research articles, Boundary Value Problems will publish review articles.
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