Multi-bump bound states for a Schrödinger system via Lyapunov–Schmidt Reduction

Abstract

Motivated by problems arising in nonlinear optics and Bose–Einstein condensates, we consider in $$\mathbb R^N$$ ( $$N \le 3$$ ) the following $$n \times n$$ system of coupled Schrodinger equations where $$\varepsilon >0$$ is a parameter, $$\beta _{ij}$$ are constants satisfying $$\beta _{ii} > 0$$ , and $$V_i$$ are positive potentials that admit some common critical points $$a_1, \ldots ,a_k$$ satisfying certain non-degenerate assumption. Then for any subsets $$J\subset \{1,2,\ldots ,k\}$$ , using a Lyapunov–Schmidt reduction method, we prove the existence of multi-bump bound solutions which as $$\varepsilon \rightarrow 0$$ concentrate on $$\cup _{j\in J}a_j$$ .