Standing waves for a system of nonlinear Schrödinger equations in \(\mathbb {R}^{N}\). (English) Zbl 1240.35155

Gloss, Elisandra

Differ. Integral Equ. 24, No. 3-4, 281-306 (2011).

Summary: We study existence and concentration of positive solutions for systems of the form \(-\varepsilon ^2\Delta u_{j}+V(x)u _{j}=F_{u_{j}}(u_1,\dots ,u_{k})\) , \(j=1,\dots ,k\) in \(\mathbb {R}^{n}\), \(u_{j}(x)\rightarrow 0\) as \(| x| \rightarrow \infty \), where \(\varepsilon >0\) is a small parameter, \(F\: [0, \infty )^{k}\rightarrow \mathbb {R}\) is a \(\mathcal {C}^{1,\alpha }_{\operatorname {loc}}\)-function, \(N\geq 3\), \(k\geq 1\), and the potential \(V\) has a positive infimum and a well.

MSC:

35J57	Boundary value problems for second-order elliptic systems
35B25	Singular perturbations in context of PDEs
35J50	Variational methods for elliptic systems
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

system of nonlinear Schrödinger equations; positive solution; potential