Let 0 < α, β < n and f, g ∈ C([0, ∞) × [0, ∞)) be two nonnegative functions. We study nonnegative classical solutions of the system

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 \({{f({\mu ^{n - \alpha }}s,{\mu ^{n - \beta }}t)} \over {{\mu ^{n + \alpha }}}}\), \({{g({\mu ^{n - \alpha }}s,{\mu ^{n - \beta }}t)} \over {{\mu ^{n + \beta }}}}\) are nonincreasing in μ > 0 for all s, t ≥ 0. 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.

设 0 < α, β < n和f, g ∈ C ([0, ∞) × [0, ∞)) 是两个非负函数。我们研究系统的非负经典解

和相应的等效积分系统。当f ( s, t ) 在s 中不减少且在t 中增加时，我们对所有此类解进行分类，g ( s, t ) 在s 中增加且在t 中不减少，并且\({{f({\mu ^{n - \alpha }}s,{\mu ^{n - \beta }}t)} \over {{\mu ^{n + \alpha }}}}\) , \({{g({\mu ^{ n - \alpha }}s,{\mu ^{n - \beta }}t)} \over {{\mu ^{n + \beta }}}}\)在μ > 0 对于所有s 不增加，吨≥ 0。我们使用的主要技术是以积分形式移动球体的方法。由于我们的假设比以前的文献中的假设更普遍，因此引入了一些新的想法来克服这个困难。