On Non-Linearly Coupled Schrodinger Systems

Abstract

Abstract. In this paper we study the following nonlinear Schr\"{o}dinger system:

\[
\begin{cases} 
-\Delta u + a(x)  u = \lambda(x)  f(u, v), & x \in \mathbb{R}^N, \\ 
-\Delta v + b(x)  v = \lambda(x)  g(u, v), & x \in \mathbb{R}^N, \\ 
u(x) \to 0, & v(x) \to 0 
\end{cases}
\]

as $ |x| \to \infty $.

Here, $ a, b, \lambda \in C(\mathbb{R}^N, \mathbb{R}) $ are all non-periodic in $ x_i $ for $ i = 1, \cdots, N, N \geq 3, f, g \in C(\mathbb{R}^2, \mathbb{R}) $. We show that this system has infinitely many solutions with small negative energies and infinitely many large-energy solutions. To the best of our knowledge, there is no corresponding result about such a Schr\"{o}dinger system.