Standing waves for coupled nonlinear Schrödinger equations with decaying potentials

Abstract

We study the following singularly perturbed problem for a coupled nonlinear Schrödinger system which arises in Bose-Einstein condensate:Here, a, b are nonnegative continuous potentials, and µ 1 , µ 2 > 0. We consider the case where the coupling constant β > 0 is relatively large. Then for sufficiently small ε > 0, we obtain positive solutions of this system which concentrate around local minima of the potentials as ε → 0. The novelty is that the potentials a and b may vanish at someplace and decay to 0 at infinity.

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

“…Our existence of positive ground state result in Theorem 1.2 is different from the one in [27] from several aspects. The result in [27] imposes weaker conditions on P , Q outside of open subset and considers positive ground state solutions concentrating near x 0 ∈ where the ground state energy m(x 0 ) for (1.3) with (λ 1 , λ 2 ) = (P (x 0 ), Q(x 0 )) achieves the minimum (see (1.7) and (1.8)).…”

“…Our result considers positive ground state solutions concentrating near global minima points of P and Q (which necessarily achieves minimal m(x 0 )) but with simplified conditions on P , Q as m(x 0 ) in (1.7) cannot be explicitly expressed. Our result holds for 1 ≤ N ≤ 3 and the one in [27] is only for N = 3 as they used Hardy inequality for the proof. Our condition (1.9) on β reveals the dependence on the spatial dimension N , and it is almost optimal compared with earlier result for autonomous system (see Theorem 1.1 and [23]).…”

“…The result in [27] imposes weaker conditions on P , Q outside of open subset and considers positive ground state solutions concentrating near x 0 ∈ where the ground state energy m(x 0 ) for (1.3) with (λ 1 , λ 2 ) = (P (x 0 ), Q(x 0 )) achieves the minimum (see (1.7) and (1.8)). Our result considers positive ground state solutions concentrating near global minima points of P and Q (which necessarily achieves minimal m(x 0 )) but with simplified conditions on P , Q as m(x 0 ) in (1.7) cannot be explicitly expressed.…”

“…For the large β > 0 case, Chen and Zou [27] proved the existence of positive solutions for (A ε ) which concentrate around local minima of the potentials for the case that P , Q may vanish in R 3 and large β. More precisely, they assume that there exist a bounded open subset of R 3 which is similar to the one in [13] so that (1.7) holds, and…”

“…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.…”