Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On elliptic systems with Sobolev critical exponent

Pages: 3357 - 3373, Volume 36, Issue 6, June 2016      doi:10.3934/dcds.2016.36.3357

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Yanfang Peng - Department of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China (email)

Abstract: We study the following elliptic system with Sobolev critical exponent \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=|u|^{2^*-2}u + \frac{\lambda\alpha}{2^*}|u|^{\alpha-2}|v|^{\beta}u,\, &x\in \mathbb{R}^N, \\ -\Delta v=|v|^{2^*-2}v + \frac{\lambda\beta}{2^*}|u|^{\alpha}|v|^{\beta-2}v,\, &x\in \mathbb{R}^N, \end{array} \right. \end{equation*} where $\lambda>0$ is a parameter, $N\geq 3$, $\alpha, \beta>1,$ $\alpha+\beta=2^*:=\frac{2N}{N-2}$, the critical Sobolev exponent. We obtain a uniqueness result on the least energy solutions and show that a manifold of a type of positive solutions is non-degenerate in some ranges of $\lambda,\alpha,\beta,N$ for the above system.

Keywords:  Positive solutions, non-degeneracy, uniqueness, system, least energy solutions.
Mathematics Subject Classification:  Primary: 35J20, 35J47; Secondary: 35J60.

Received: October 2014;      Revised: October 2015;      Available Online: December 2015.