Positive vector solutions for a Schrödinger system with external source terms

Abstract

This paper is concerned with the existence of many synchronized vector solutions for the following Schrodinger system with external source terms $$\begin{aligned} \left\{ \begin{array}{ll} - \Delta u + u=a(x) u^3+\beta u v^2+f(x), &{}\quad x\in {\mathbb {R}}^3,\\ - \Delta v + v=b(x) v^3+\beta u^2 v+g(x), &{}\quad x\in {\mathbb {R}}^3,\\ u,v >0, &{}\quad x\in {\mathbb {R}}^3, \end{array} \right. \end{aligned}$$where $$\beta \in {\mathbb {R}}$$ is a coupling constant, $$a, b\in C({\mathbb {R}}^3)$$ and $$f,\ g\in L^2({\mathbb {R}}^3)\cap L^\infty ({\mathbb {R}}^3)$$. This type of system arises in Bose–Einstein condensates and Kerr-like photo refractive media. This paper tries to reveal the influence of the external source terms f and g on the number of the solutions. It is shown that the level set of the corresponding functional has a quite rich topology and the system admits k spikes synchronized vector solutions for any $$k\in \mathbb {Z}^+$$ when f and g are small and a(x), b(x) satisfy some additional assumptions at infinity. The proof is based on the Lyapunov–Schmidt reduction scheme and the main ingredient is to improve the estimate on the remainder term obtained in the reduction process.