Existence of solutions to higher order Lane-Emden type systems

Abstract

We prove existence results for the Lane-Emden type system \[ \begin{cases} \begin{aligned} (-Δ)^α u=\left| v \right|^q \\ (-Δ)^β v= \left| u \right|^p \end{aligned} \text{ in } B_1 \subset \mathbb{R}^N \\ \frac{\partial^{r} u}{\partial ν^{r}}=0, \, r=0, \dots, α-1, \text{ on } \partial B_1 \\ \frac{\partial^{r} v}{\partial ν^{r}}=0, \, r=0, \dots, β-1, \text{ on } \partial B_1. \end{cases} \] where $B_1$ is the unitary ball in $\mathbb{R}^N$, $N >\max \{2α, 2β\}$, $ν$ is the outward pointing normal, $α, β\in \mathbb{N}$, $α, β\ge 1$ and $(-Δ)^α= -Δ((-Δ)^{α-1})$ is the polyharmonic operator. A continuation method together with a priori estimates will be exploited. Moreover, we prove uniqueness for the particular case $α=2$, $β=1$ and $p, q>1$.