Archiv der Mathematik ( IF 0.6 ) Pub Date : 2020-10-01 , DOI: 10.1007/s00013-020-01519-3 Zhipeng Yang , Yuanyang Yu
In this paper, we study the following nonlinear elliptic systems:
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1+V_1(x)u_1=\partial _{u_1}F(x,u)&{}\quad x\in {\mathbb {R}}^N,\\ -\Delta u_2+V_2(x)u_2=\partial _{u_2}F(x,u)&{}\quad x\in {\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$where \(u=(u_1,u_2):{\mathbb {R}}^N\rightarrow {\mathbb {R}}^2\), F and \(V_i\) are periodic in \(x_1,\ldots ,x_N\) and \(0\notin \sigma (-\,\Delta +V_i)\) for \(i=1,2\), where \(\sigma (-\,\Delta +V_i)\) stands for the spectrum of the Schrödinger operator \(-\,\Delta +V_i\). Under some suitable assumptions on F and \(V_i\), we obtain the existence of infinitely many geometrically distinct solutions. The result presented in this paper generalizes the result in Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009).
中文翻译：
具有周期势的非线性椭圆系统的几何上不同的解
在本文中，我们研究以下非线性椭圆系统：
$$ \ begin {aligned} {\ left \ {\ begin {array} {ll}-\ Delta u_1 + V_1（x）u_1 = \ partial _ {u_1} F（x，u）＆{} \ quad x \在{\ mathbb {R}} ^ N中，\\-\ Delta u_2 + V_2（x）u_2 = \ partial _ {u_2} F（x，u）＆{} \ quad x \在{\ mathbb {R}中} ^ N，\ end {array} \右。} \ end {aligned} $$其中\（u =（u_1，u_2）：{\ mathbb {R}} ^ N \ rightarrow {\ mathbb {R}} ^ 2 \），F和\（V_i \）在\（x_1，\ ldots中是周期性的，x_N \）和\（0 \ notin \ sigma（-\，\ Delta + V_i）\）表示\（i = 1,2 \），其中\（\ sigma（-\，\ Delta + V_i）\）代表Schrödinger运算符\（-\，\ Delta + V_i \）的频谱。在F和\（V_i \）的一些适当假设下，我们获得了无限多个几何上不同的解的存在。本文介绍的结果将结果推广到Szulkin和Weth（J Funct Anal 257（12）：3802-3822，2009）。
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