The Positive Solutions To Integral Equation (System) Involving Wolff Potential

Many researchers have studied the integral equations(system) involving Wolff po-tential in Rn. But most of their results are not weighted integral equation. What’s more,in this paper, we consider the weighted integral equation (system) involving Wolff po-tential.In this paper, we mainly study some properties of the positive solutions of equa-tion(system), such as integrability, boundedness, symmetry, decay rates etc. We use themethod of regularity lifting the method of diagonal extending to obtain the integrabilityof the solutions and show that the integrability intervals are optimal. In addition, weclassify the ground states of the solutions of equation(system). Namely, we prove thatthe solutions are bounded and they converge to zero when|x|→0.In this paper, we assume R(x), R1(x), R2(x) are bounded. When they are notconstants, the solutions of integral equation(system) involving them have no radialstructure. About the symmetry, we mainly consider the case of the three boundedfunction are always equivalent to1. By using the method of moving planes, we showthat the solutions are radially symmetric and decreasing about some x0∈Rn. At last,we obtain the decay rates of the solutions near infinity in this case.The paper consists of three chapters:In Chapter1, we introduce the sciential background, having obtained results, themain works in this paper and its prepare knowledge of this paper;In Chapter2, we mainly discuss some properties of the positive solutions of theintegral equation involving Wolff potential in Rn. In addition, we classify the groundstates of the solutions of the which is closely with the equation.In Chapter3, we mainly discuss some properties of the positive solutions of theintegral system involving Wolff potential in Rn.