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 Schrödinger equations where %$\varepsilon >0%$ is a parameter, %$\beta _{ij}%$ are constants satisfying %$\beta _{i...
Ausführliche Beschreibung

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 Schrödinger 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 %$\cup _{j\in J}a_j%$. Ausführliche Beschreibung