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), p. 429-447 / Harvested from Numdam

Résumé

We study the following linearly coupled Schrödinger equations: ${\begin{matrix} - ϵ^{2} Δ u + a (x) u = u^{p} + λ v, x \in ℝ^{N}, \\ - ϵ^{2} Δ v + b (x) v = v^{2^{⁎} - 1} + λ u, x \in ℝ^{N}, \\ u, v > 0 in ℝ^{N}, u (x), v (x) \to 0 as | x | \to \infty, \end{matrix}$ where $N ⩾ 3$ , $2^{⁎} = \frac{2 N}{N - 2}$ , $1 < p < 2^{⁎} - 1$ , and $a (x), b (x)$ are positive continuous potentials which are both bounded away from 0. Under some assumptions on $a (x)$ and $λ > 0$ , 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 $b (x)$ .

Publié le : 2014-01-01

Zbl 1300.35029

DOI : https://doi.org/10.1016/j.anihpc.2013.04.003

@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},
     volume = {31},
     year = {2014},
     pages = {429-447},
     doi = {10.1016/j.anihpc.2013.04.003},
     zbl = {1300.35029},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 31 (2014) pp. 429-447. doi : 10.1016/j.anihpc.2013.04.003. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_3_429_0/

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