AXIAL SYMMETRY AND REGULARITY OF SOLUTIONS TO AN INTEGRAL EQUATION IN A HALF-SPACE

摘要

We consider the integral equation u(x) = ∫_(R~n) G(x, y) f (u(y)) d y, where G(x, y) is the Green's function of the corresponding polyharmonic Dirichlet problem in a half-space. We prove by the method of moving planes in integral form that, under some integrability conditions, the solutions are axially symmetric with respect to some line parallel to the x_n -axis and non-decreasing in the x_n direction, which further implies the nonexistence of solutions. We also show similar results for a class of systems of integral equations. This appears to be the first paper in which the moving plane method in integral form is employed in a half-space to derive axial symmetry. We also obtain the regularity of the integral equation in a half-space u(x) ∫_(R~n) G(x,y)|u(y)|~(p-1_u(y)dy by the regularity lifting method. As a corollary, we prove the nonexistence of nonnegative solutions to this equation. Moreover, we show that the non-negative solutions in this equation only depend on x_n if u ∈ L_(loc)~(2n/(n-2m)) (R_+~-) and 1 < p < (n 2m)/ (n-2m).