Classification of Solutions for a System of Integral Equations

Symmetry of solutions to some systems of integral equations

“…Remark 1.5. This theorem is a generalization of the results in [9] and [8] about the classification of nonnegative solutions.…”

“…The integrability conditions ensure that one can choose sufficiently small, so that for all κ in [κ o , κ o + ), Another application of Theorem 1.2 is the classification of the system (1.4), which has been discussed thoroughly in [9]. In this system, the integrability conditions are u ∈ L p+1 (R n ) and v ∈ L q+1 (R n ).…”

“…Furthermore, another paper from Chen, Li, an Ou [9] discussed the actual system of integral equations that maximize the constant in the Hardy-Littlewood-Sobolev inequality. They presented and proved:…”

The best constant in a weighted Hardy-Littlewood-Sobolev inequality

“…To obtain the best constant in the weighted inequality (3), one can maximize the functional (5) J(f, g) = R n R n f (x)g(y) |x| α |x − y| λ |y| β dxdy under the constraints f r = g s = 1. The corresponding Euler-Lagrange equations are the system of integral equations:…”

Liouville-Type Theorems for Some Integral Systems

“…In this paper, by virtue of Hardy-Littlewood-Sololev inequality instead of Maximum Principle, we consider the integral systems (1) and (2) with N    0 and the general nonlinearities. Therefore, it is a generalization of Liouville-type theorems in [5,6,8,9,[13][14][15].…”