Existence of solutions to higher order Lane-Emden type systems

Abstract: We prove existence results for the Lane-Emden type system [ \begin{cases} \begin{aligned} (-\Delta)^{\alpha} u=\left| v \right|^q \ (-\Delta)^{\beta} v= \left| u \right|^p \end{aligned} \text{ in } B_1 \subset \mathbb{R}^N \ \frac{\partial^{r} u}{\partial \nu^{r}}=0, \, r=0, \dots, \alpha-1, \text{ on } \partial B_1 \ \frac{\partial^{r} v}{\partial \nu^{r}}=0, \, r=0, \dots, \beta-1, \text{ on } \partial B_1. \end{cases} ] where $B_1$ is the unitary ball in $\mathbb{R}^N$, $N >\max {2\alpha, 2\beta }$, $\nu$ is the outward pointing normal, $\alpha, \beta \in \mathbb{N}$, $\alpha, \beta \ge 1$ 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 $p, q>1$.