Existence and asymptotic behavior of solitary waves for a weakly coupled Schrödinger system

Abstract

This paper deals with the following weakly coupled nonlinear Schrödinger system −Δu1+a1(x)u1=∣u1∣2p−2u1+b∣u1∣p−2∣u2∣pu1,x∈RN,−Δu2+a2(x)u2=∣u2∣2p−2u2+b∣u2∣p−2∣u1∣pu2,x∈RN,\left\{\begin{array}{ll}-\Delta {u}_{1}+{a}_{1}\left(x){u}_{1}=| {u}_{1}{| }^{2p-2}{u}_{1}+b| {u}_{1}{| }^{p-2}| {u}_{2}{| }^{p}{u}_{1},& x\in {{\mathbb{R}}}^{N},\\ -\Delta {u}_{2}+{a}_{2}\left(x){u}_{2}=| {u}_{2}{| }^{2p-2}{u}_{2}+b| {u}_{2}{| }^{p-2}| {u}_{1}{| }^{p}{u}_{2},& x\in {{\mathbb{R}}}^{N},\end{array}\right. where N≥1N\ge 1, b∈Rb\in {\mathbb{R}} is a coupling constant, 2p∈(2,2∗)2p\in \left(2,{2}^{\ast }), 2∗=2N/(N−2){2}^{\ast }=2N\hspace{0.1em}\text{/}\hspace{0.1em}\left(N-2) if N≥3N\ge 3 and +∞+\infty if N=1,2N=1,2, a1(x){a}_{1}\left(x) and a2(x){a}_{2}\left(x) are two positive functions. Assuming that ai(x)(i=1,2){a}_{i}\left(x)\hspace{0.33em}\left(i=1,2) satisfies some suitable conditions, by constructing creatively two types of two-dimensional mountain-pass geometries, we obtain a positive synchronized solution for ∣b∣>0| b| \gt 0 small and a positive segregated solution for b0b\gt 0 is small. The asymptotic behavior of the solutions when b→0b\to 0 and b→−∞b\to -\infty is also studied.

Keywords

• bose-einstein condensates
• nonlinear optics
• nonunique
• nontrivial constraints
• schrödinger systems
• segregated
• synchronized
• weakly coupled
• 35b50
• 35j10
• 35j50