Standing waves for linearly coupled Schrödinger equations with critical exponent
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, p. 429-447
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We study the following linearly coupled Schrödinger equations: $\left\{\begin{array}{c}-{ϵ}^{2}\Delta u+a\left(x\right)u={u}^{p}+\lambda v,\phantom{\rule{1em}{0ex}}x\in {ℝ}^{N},\hfill \\ -{ϵ}^{2}\Delta v+b\left(x\right)v={v}^{{2}^{⁎}-1}+\lambda u,\phantom{\rule{1em}{0ex}}x\in {ℝ}^{N},\hfill \\ u,v>0\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}{ℝ}^{N},\phantom{\rule{1em}{0ex}}u\left(x\right),v\left(x\right)\to 0\phantom{\rule{1em}{0ex}}\text{as}\phantom{\rule{4pt}{0ex}}|x|\to \infty ,\hfill \end{array}$ where $N⩾3$, ${2}^{⁎}=\frac{2N}{N-2}$, $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 $b\left(x\right)$.
@article{AIHPC_2014__31_3_429_0,
author = {Chen, Zhijie and Zou, Wenming},
title = {Standing waves for linearly coupled Schr\"odinger equations with critical exponent},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {31},
number = {3},
year = {2014},
pages = {429-447},
doi = {10.1016/j.anihpc.2013.04.003},
zbl = {1300.35029},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2014__31_3_429_0}
}

Chen, Zhijie; Zou, Wenming. Standing waves for linearly coupled Schrödinger equations with critical exponent. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, pp. 429-447. doi : 10.1016/j.anihpc.2013.04.003. http://www.numdam.org/item/AIHPC_2014__31_3_429_0/

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