Existence of solutions to higher order Lane-Emden type systems

We prove existence results for the Lane-Emden type system $$\{\begin{array}{l}\begin{array}{r}{\displaystyle (-\mathrm{\Delta }{)}^{\alpha }u={\mid v\mid }^q}\\ {\displaystyle (-\mathrm{\Delta }{)}^{\beta }v={\mid u\mid }^p}\end{array}\text{ in }{B}_1\subset {\mathbb{R}}^N\\ \frac{{\mathrm{\partial }}^{r}u}{\mathrm{\partial }{\nu }^r}=0,r=0,\dots ,\alpha -1,\text{ on }\mathrm{\partial }{B}_1\\ \frac{{\mathrm{\partial }}^{r}v}{\mathrm{\partial }{\nu }^r}=0,r=0,\dots ,\beta -1,\text{ on }\mathrm{\partial }{B}_1\mathrm{.}\end{array}$$ where $B_1$ is the unitary ball in ${\mathbb{R}}^N$, $N >\max {2\alpha, 2\beta }$,$\nu$istheoutwardpointingnormal,$\alpha, \beta \in $\mathbb{N}$$,$\alpha, \beta \ge 1$and$(-\Delta)^{\alpha}= -\Delta((-\Delta)^{\alpha-1})$ is the polyharmonic operator. A continuation method together with a priori estimates will be exploited. Moreover, we prove uniqueness for the particular case $\alpha =2$, $\beta =1$ and $p,q>1$.