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

作者：Lu Guozhen^*; Zhu Jiuyi

来源：Pacific Journal of Mathematics, 2011, 253(2): 455-473.

• axial symmetry
• regularity of solutions
• half-space
• Green's functions for polyharmonic operators
• integral equation
• nonexistence of solutions

We consider the integral equation
u(x) = integral(Rn+) G(x, y) f(u(y)) dy,
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) = integral(Rn+) G(x, y) f (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 nonnegative solutions in this equation only depend on x(n) if u is an element of L-loc(2n/(n-2m)) (R-+(n)) and 1 < p < (n + 2m)/(n - 2m).

• 出版日期2011-10