A local mountain pass type result for a system of nonlinear Schrödinger equations

Abstract

We consider a singular perturbation problem for a system of nonlinear Schrödinger equations:where N = 2, 3, μ 1 , μ 2 , β > 0 and V 1 (x), V 2 (x) : R N → (0, ∞) are positive continuous functions. We consider the case where the interaction β > 0 is relatively small and we define for P ∈ 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 setWe show that ( * ) possesses a family of non-trivial vector positive solutions {(v 1ε (x), v 2ε (x))} ε∈(0,ε 0 ] which concentrates-after extracting a subsequence ε n → 0-to a point P 0 ∈ Λ Communicated by P. Rabinowitz. 123 450 N. Ikoma, K. Tanaka with m(P 0 ) = inf P∈Λ m(P). Moreover (v 1ε (x), v 2ε (x)) converges to a least energy nontrivial vector solution of ( * * ) after a suitable rescaling. Mathematics Subject Classification (2000)35B25 · 35J65 · 58E05

Standing waves of a weakly coupled Schrödinger system with distinct potential functions

“…The existence, multiplicity, bifurcation, concentration behavior of positive solutions of (A ε ) with ε = 1 and (1.3) (in R N or a bounded domain of R N ) have been considered in, for example, [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and references therein. In particular for the autonomous case (1.3), the existence, uniqueness of positive solutions can be summarized as follows: Theorem 1.1.…”

“…The semiclassical case (A ε ) with trapping potentials P (x) and Q(x) has been studied in [27,13,16,20,21]. Lin and Wei [16] proved the existence and asymptotic concentration behavior of a ground state solution of (A ε ) with −∞ < β < β 0 for a small β 0 > 0.…”

Semi-classical standing waves for nonlinear Schrödinger systems

“…The difficulty is overcome by getting a lower bound (∇ y L , ∇ u L) in a tubular neighborhood of a set of approximate solutions and using the equi-continuity of the energy gradient flow and the translation flow in a neighborhood of a set of approximate solutions for small ε > 0. There are some previous results [21,27,29] for vector solutions of (3) and [26,31] for vector solutions of (3) on a bounded domain with the Dirichlet boundary condition. Lin and Wei in [27] studied the existence and asymptotic behavior of a least energy vector solution of (3) for trap potentials W 1 , W 2 for a β in a certain range; each components of the vector solutions may have common or different concentration points depending on the parameter β and the shape of potentials W , W 2 .…”

“…Lin and Wei in [27] studied the existence and asymptotic behavior of a least energy vector solution of (3) for trap potentials W 1 , W 2 for a β in a certain range; each components of the vector solutions may have common or different concentration points depending on the parameter β and the shape of potentials W , W 2 . Ikoma and Tanaka in [21] proved nicely the existence of a vector solution with a common concentration point of each components in the setting of local mountain pass for vector solutions when β ∈ (0, min{ √ μ 1 μ 2 , β * }), where β * is a local constant defined by linearized eigenvalues β 1 , β 2 in (11) and (12). (11) and (12), defining…”

“…This implies that Morse index of the vector solution is 2 for β ∈ (0, min{β 1 , β 2 }) and 1 for β ∈ (max{β 1 , β 2 }, ∞). It is known in [21] that for β ∈ (0, min{β 1 , β 2 }), the vector solution U y is a least energy solution among all vector solutions of (5) if an additional condition β < √ μ 1 μ 2 is satisfied. When W 1 (y) = W 2 (y), it is known that {μ 1 , μ 2 } = {β 1 , β 2 } and there exist no vector solutions for β ∈ (min{β 1 , β 2 }, max{β 1 , β 2 }).…”