Existence of solutions to higher order Lane-Emden type systems

TL;DR

This paper establishes the existence of solutions for a higher order Lane-Emden system involving polyharmonic operators in a bounded domain, using continuation methods and a priori estimates, and proves uniqueness in a specific case.

Contribution

It provides new existence results for higher order Lane-Emden systems with boundary conditions, extending previous work to polyharmonic operators and including a uniqueness result for a special case.

Findings

Existence of solutions for the system in the unit ball.

Use of continuation method and a priori estimates.

Uniqueness proven for the case d7=2, d7=1, p, q > 1.

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 $p, q>1$.