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

Permanent link to this document
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