Abstract and Applied Analysis

Multiplicity of Positive Solutions for Semilinear Elliptic Systems

Tsing-San Hsu

Abstract

We study the effect of the coefficient $h\left(x\right)$ of the critical nonlinearity on the number of positive solutions for semilinear elliptic systems. Under suitable assumptions for $f\left(x\right)$, $g\left(x\right)$, and $h\left(x\right)$, we should prove that for sufficiently small $\lambda ,\mathrm{}\mu >\mathrm{0}$, there are at least $k+\mathrm{1}$ positive solutions of the semilinear elliptic systems $-\mathrm{\Delta }u=\lambda f\left(x\right)|u{|}^{q-\mathrm{2}}u+\mathrm{\left(\alpha }/\left(\alpha +\mathrm{\beta \right)}\right)h\left(x\right)|u{|}^{\alpha -\mathrm{2}}u|v{|}^{\beta }$, $\mathrm{}-\mathrm{\Delta }v=\mu g\left(x\right)|v{|}^{q-\mathrm{2}}v+\mathrm{\left(\beta }/\left(\alpha +\mathrm{\beta \right)}\right)h\left(x\right)|u{|}^{\alpha }|v{|}^{\beta -\mathrm{2}}v$, where $\mathrm{0}\in \mathrm{\Omega }\subset {ℝ}^{N}$ is a bounded domain, $\alpha >\mathrm{1}$, $\beta >\mathrm{1}$, and $N/\left(N-\mathrm{2}\right) for $N>\mathrm{4}$.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 746380, 13 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511842

Digital Object Identifier
doi:10.1155/2013/746380

Mathematical Reviews number (MathSciNet)
MR3044999

Zentralblatt MATH identifier
1277.35161

Citation

Hsu, Tsing-San. Multiplicity of Positive Solutions for Semilinear Elliptic Systems. Abstr. Appl. Anal. 2013 (2013), Article ID 746380, 13 pages. doi:10.1155/2013/746380. https://projecteuclid.org/euclid.aaa/1393511842

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