Electron. J. Diff. Equ., Vol. 2013 (2013), No. 15, pp. 1-14.

Multiple positive solutions for quasilinear elliptic systems

Qin Li, Zuodong Yang

Abstract:
In this article, we investigate how the coefficient $f(z)$ affects the number of positive solutions of the quasilinear elliptic system
$$\displaylines{ -\Delta_{p}u =\lambda g(z)|u|^{q-2}u+\frac{\alpha}{\alpha+\beta} f(z)|u|^{\alpha-2}u|v|^{\beta} \quad\hbox{in }\Omega,\cr -\Delta_{p}v =\mu h(z)|v|^{q-2}v +\frac{\beta}{\alpha+\beta}f(z)|u|^{\alpha}|v|^{\beta-2}v \quad\hbox{in }\Omega,\cr u=v=0\quad\hbox{on }\partial\Omega, }$$
where $0\in\Omega\subset \mathbb{R}^{N}$ is a bounded domain, $\alpha >1$, $\beta>1$ and $1<p<q<\alpha+\beta=p^{*}=\frac{Np}{N-p}$ for $N>2p$.

Submitted July 6, 2012. Published January 17, 2013.
Math Subject Classifications: 35J65, 35J50.
Key Words: Quasilinear elliptic systems; multiple positive solutions; critical point, Nehari manifold.

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  Qin Li
Institute of Mathematics, School of Mathematical Sciences
Nanjing Normal University
Jiangsu Nanjing, China
email: 294973245@qq.com
Zuodong Yang
Institute of Mathematics, School of Mathematical Sciences
Nanjing Normal University
Jiangsu Nanjing, China
email: zdyang_jin@263.net

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