Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (4): 1302-1320.

• Articles •

### CLASSIFICATION OF SOLUTIONS TO HIGHER FRACTIONAL ORDER SYSTEMS

Phuong LE

1. Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam Vietnam National University, Ho Chi Minh City, Vietnam
• Received:2020-07-14 Revised:2020-10-01 Online:2021-08-25 Published:2021-09-01
• Supported by:
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2020.22.

Abstract: Let $0 < \alpha,\beta < n$ and $f,g \in C([0,\infty) \times[0,\infty))$ be two nonnegative functions. We study nonnegative classical solutions of the system $\begin{cases} (-\Delta)^{\frac{\alpha}{2}} u=f(u,v) &\text{ in } \mathbb{R}^n,\\ (-\Delta)^{\frac{\beta}{2}} v=g(u,v) &\text{ in } \mathbb{R}^n, \end{cases}$ and the corresponding equivalent integral system. We classify all such solutions when $f(s,t)$ is nondecreasing in $s$ and increasing in $t$, $g(s,t)$ is increasing in $s$ and nondecreasing in $t$, and $\frac{f(\mu^{n-\alpha}s,\mu^{n-\beta}t)}{\mu^{n+\alpha}}$, $\frac{g(\mu^{n-\alpha}s,\mu^{n-\beta}t)}{\mu^{n+\beta}}$ are nonincreasing in $\mu > 0$ for all $s,t\ge0$. The main technique we use is the method of moving spheres in integral forms. Since our assumptions are more general than those in the previous literature, some new ideas are introduced to overcome this difficulty.

CLC Number:

• 45G15
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