TITLE

# A local mountain pass type result for a system of nonlinear Schrï¿½dinger equations

AUTHOR(S)
Ikoma, Norihisa; Tanaka, Kazunaga
PUB. DATE
March 2011
SOURCE
Calculus of Variations & Partial Differential Equations;Mar2011, Vol. 40 Issue 3/4, p449
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
We consider a singular perturbation problem for a system of nonlinear Schrï¿½dinger equations: where N = 2, 3, ï¿½, ï¿½, ï¿½ > 0 and V( x), V( x): R ? (0, 8) are positive continuous functions. We consider the case where the interaction ï¿½ > 0 is relatively small and we define for $${P\in{\bf R}^N}$$ the least energy level m( P) for non-trivial vector solutions of the rescaled 'limit' problem:We assume that there exists an open bounded set $${\Lambda\subset{\bf R}^N}$$ satisfyingWe show that (*) possesses a family of non-trivial vector positive solutions $${\{(v_{1\varepsilon}(x), v_{2\varepsilon} (x))\}_{\varepsilon\in (0,\varepsilon_0]}}$$ which concentrates-after extracting a subsequence e ? 0-to a point $${P_0\in\Lambda}$$ with $${m(P_0)={\rm inf}_{P\in\Lambda}m(P)}$$. Moreover ( v( x), v( x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.
ACCESSION #
57389810

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