Qualitative properties for some integral systems involving Riesz potentials

Abstract

This paper is concerned with the following integral system involving Riesz potentials: $\{\begin{array}{l}{\displaystyle u\left(x\right)={\int }_{R^n}{\displaystyle \frac{v^q\left(y\right)}{|x-y{|}^{n-\alpha }}}\mathrm{d}y,u>0\mathrm{i}\mathrm{n}{R}^n,}\\ {\displaystyle v\left(x\right)={\int }_{R^n}{\displaystyle \frac{u^p\left(y\right)}{|x-y{|}^{n-\beta }}}\mathrm{d}y,v>0\mathrm{i}\mathrm{n}{R}^n,}\end{array}$ where $p,q>1,0<\alpha ,\text{β}<n.$ It can be regarded as a generalization of the well-known Hardy–Littlewood–Sobolev system, but it has some difficulties. To overcome this, we employ the regularity lifting lemma and classified discussion to obtain the optimal integrable intervals of the positive solutions. Combining with the radial symmetry and monotonicity, the decay rates at infinity of the solutions are established. These properties are the key ingredients to understand the positive solutions.

Keywords:

• Riesz potentials
• regularity liftings
• decay rates

Mathematics Subject Classifications:

• 45G15
• 45M05
• 47N20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the National Natural Science Foundation of China (NNSF) [11871278].