Multi-Peak Solutions for Coupled Nonlinear Schrödinger Systems in Low Dimensions.

In this paper, we construct the solutions to the following nonlinear Schrödinger system - ϵ 2 Δ u + P (x) u = μ 1 u p + β u p - 1 2 v p + 1 2 in R N , - ϵ 2 Δ v + Q (x) v = μ 2 v p + β u p + 1 2 v p - 1 2 in R N , where 3 < p < + ∞ , N ∈ { 1 , 2 } , ϵ > 0 is a small parameter, the potentials P, Q satisfy 0 < P 0 ≤ P (x) ≤ P 1 and Q(x) satisfies 0 < Q 0 ≤ Q (x) ≤ Q 1 . We construct the solution for attractive and repulsive cases. When x 0 is a local maximum point of the potentials P and Q and P (x 0) = Q (x 0) , we construct k spikes concentrating near the local maximum point x 0 . When x 0 is a local maximum point of P and x ¯ 0 is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point x 0 and m spikes of v concentrating at the local maximum point x ¯ 0 when x 0 ≠ x ¯ 0. This paper extends the main results established by Peng and Wang (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case N = 3 , p = 3 . [ABSTRACT FROM AUTHOR]