## Existence of solutions for a modified nonlinear Schrödinger system.(English)Zbl 1271.35065

Summary: We are concerned with the following modified nonlinear Schrödinger system: $$-\Delta u + u - (1/2)u\Delta(u^2) = (2\alpha/(\alpha + \beta))|u|^{\alpha - 2}|v|^\beta u$$, $$x \in \Omega$$, $$-\Delta v + v - (1/2)v\Delta(v^2) = (2\beta/(\alpha + \beta))|u|^\alpha|v|^{\beta - 2}v$$, $$x \in \Omega$$, $$u = 0$$, $$v = 0$$, $$x \in \partial \Omega$$, where $$\alpha > 2$$, $$\beta > 2$$, $$\alpha + \beta < 2 \cdot 2^\ast$$, $$2^\ast = 2N/(N - 2)$$ is the critical Sobolev exponent, and $$\Omega \subset \mathbb R^N (N \geq 3)$$ is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B09 Positive solutions to PDEs 35B20 Perturbations in context of PDEs
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### References:

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