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 ${\mathbb{R}}^N$, $N>\max \{2\alpha ,2\beta \}$, $\nu$ 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 $p,q>1$.

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• Uniqueness results for higher order elliptic equations and systems