The Structure And Properties Of Solutions For A General Integral Equation

In this paper, we consider the following general integral equatione: and obtain regularity, radial symmetry, and monotonicity of the solutions of the equa-tion.We also consider the equivalence between integral equation (0-1) and the partial dif-ferential equation in the weak sense. We also obtain some corollaries of our theorems.Our main results are the following theorems:Theorem 1 Assume that u∈Lq(Rn) is a solution of (0-1) for some q>n/(n-α). Suppose thatThen u(x) is in L∞(Rn) and hence is continuous.Theorem 2 Let u∈Lq(Rn) be a solution of (0-1) for some q>n/(n-α). Assume that(ⅰ) f(x,u) and ((?)f)/((?)u) are strictly increasing in u,(ⅱ)∫Rn|((?)f)/((?)u)(y,u(y))|n/αdy<∞,and(ⅲ) f(x,u) is symmetric and decreasing about the origin in x1-direction.Then u is symmetric and decreasing about the origin in x1-direction.Corollary 1 Let u∈Lq(Rn) be a solution of (0-1) satisfyingⅰ) andⅱ) in Theorem 2. In addition, assume that f=f(|x|,u) and f is strictly decreasing in |x|, then u is radially symmetric and decreasing about the origin of Rn.Corollary 2 Let u∈Lq(Rn) be a solution of (0-1) satisfyingⅰ) andⅱ) in Theorem 2. If f= f(u) then u must be radially symmetric and decreasing about some point in Rn.Theorem 3 Every solution of (0-1) multiplied by a constant is also a weak solution of and vice versa.In section 3, we prove the regularity of the solutions by using regularity lifting and obtain symmetry and monotonicity of the solutions by using the method of moving planes. We also prove the equivalence between (0-1) and the (0-2) in the weak sense.Our proofs mainly depend on regularity of the solutions, the extremum principle of integral inequalities and the Hardy-Littlewood-Sobolev inequality.