Standing waves for linearly coupled Schrödinger equations with critical exponent

Zhijie Chen; Wenming Zou

doi:10.1016/j.anihpc.2013.04.003

JournalsaihpcVol. 31, No. 3pp. 429–447

Standing waves for linearly coupled Schrödinger equations with critical exponent

Zhijie Chen
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Wenming Zou
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

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Abstract

We study the following linearly coupled Schrödinger equations:

⎩ ⎨ ⎧ - 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,

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 $b (x)$ .

Cite this article

Zhijie Chen, Wenming Zou, Standing waves for linearly coupled Schrödinger equations with critical exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 3, pp. 429–447

DOI 10.1016/J.ANIHPC.2013.04.003