Uniqueness of Positive Bound States to Schrödinger Systems with Critical Exponents
Uniqueness of Positive Bound States to Schrödinger Systems with Critical Exponents
复制标题
DOI：
10.1137/080712301
复制
发表时间：
2007-08
期刊：
影响因子：
--
通讯作者：
Congming Li;Li Ma
中科院分区：
文献类型：
--
作者：
Congming Li;Li Ma
证明了如下椭圆型方程组：（1）$-\Delta（u（x））=u（x）^{\alpha}v（x）^{\beta}$，（2）$-\Delta（v（x））=u（x）^{\beta}v（x）^{\alpha}$正解的唯一性.这里$x\in R^n$，$n\geq3$，和$1\leq\alpha<\beta\leq\frac{n+2}{n-2}$，其中$\alpha+\beta=\frac{n+2}{n-2}$。在特殊情况下，当$n=3$和$\alpha =2$，$\beta=3$时，该系统与具有玻色-爱因斯坦凝聚临界指数的定态薛定谔系统密切相关。作为第一步，我们证明了上述具有临界指数的椭圆型方程组正解的径向对称性。然后，我们证明了$u=v$，这是我们唯一性结果的关键点。
We prove the uniqueness of the positive solutions of the following elliptic system: (1) $-\Delta(u(x))=u(x)^{\alpha}v(x)^{\beta}$, (2) $-\Delta(v(x))=u(x)^{\beta}v(x)^{\alpha}$. Here $x\in R^n$, $n\geq3$, and $1\leq\alpha<\beta\leq\frac{n+2}{n-2}$ with $\alpha+\beta=\frac{n+2}{n-2}$. In the special case when $n=3$ and $\alpha =2$, $\beta=3$, the system is closely related to the ones from the stationary Schrodinger system with critical exponents for the Bose–Einstein condensate. As the first step, we prove the radial symmetry of the positive solutions to the elliptic system above with critical exponents. We then prove that $u=v$, which is a key point for our uniqueness result.