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2013 Existence of Solutions for a Modified Nonlinear Schrödinger System
Yujuan Jiao, Yanli Wang
J. Appl. Math. 2013: 1-9 (2013). DOI: 10.1155/2013/431672

## 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.

## Citation

Yujuan Jiao. Yanli Wang. "Existence of Solutions for a Modified Nonlinear Schrödinger System." J. Appl. Math. 2013 1 - 9, 2013. https://doi.org/10.1155/2013/431672

## Information

Published: 2013
First available in Project Euclid: 14 March 2014

zbMATH: 1271.35065
MathSciNet: MR3074319
Digital Object Identifier: 10.1155/2013/431672  