Standing waves for linearly coupled Schrödinger equations with critical exponent

Chen, Zhijie; Zou, Wenming

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 3, pp. 429-447.

We study the following linearly coupled Schrödinger equations:

$$\{\begin{array}{c}-{ϵ}^2\Delta u+a\left(x\right)u=u^p+\lambda v,x\in {ℝ}^N,\hfill \\ -{ϵ}^2\Delta v+b\left(x\right)v=v^{2^{⁎}-1}+\lambda u,x\in {ℝ}^N,\hfill \\ u,v>0\text{in}{ℝ}^N,u\left(x\right),v\left(x\right)\to 0\text{as}\left|x\right|\to \infty ,\hfill \end{array}$$

where $N⩾3$, $2^{⁎}=\frac{2N}{N-2}$, $1<p<2^{⁎}-1$, and $a\left(x\right),b\left(x\right)$ are positive continuous potentials which are both bounded away from 0. Under some assumptions on $a\left(x\right)$ and $\lambda >0$, we obtain positive solutions of the coupled system for sufficiently small $ϵ>0$, which have concentration phenomenon as $ϵ\to 0$. It is interesting that we do not need any further assumptions on $b\left(x\right)$.

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