## Journal of Applied Mathematics

### Existence of Solutions for a Modified Nonlinear Schrödinger System

#### Abstract

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

#### Article information

Source
J. Appl. Math. Volume 2013 (2013), Article ID 431672, 9 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394808044

Digital Object Identifier
doi:10.1155/2013/431672

Mathematical Reviews number (MathSciNet)
MR3074319

Zentralblatt MATH identifier
1271.35065

#### Citation

Jiao, Yujuan; Wang, Yanli. Existence of Solutions for a Modified Nonlinear Schrödinger System. J. Appl. Math. 2013 (2013), Article ID 431672, 9 pages. doi:10.1155/2013/431672. https://projecteuclid.org/euclid.jam/1394808044

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