Vol. 253, No. 2, 2011

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Axial symmetry and regularity of solutions to an integral equation in a half-space

Guozhen Lu and Jiuyi Zhu

Vol. 253 (2011), No. 2, 455–473
Abstract

We consider the integral equation

       ∫
u(x) =    G (x,y)f(u(y))dy,
ℝn+

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 xn-axis and nondecreasing in the xn 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) =    G(x,y)|u(y)|p−1u(y)dy
ℝn+

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 xn if u Lloc2n∕(n2m)(+n) and 1 < p < (n + 2m)(n 2m).

Keywords
axial symmetry, regularity of solutions, half-space, Green’s functions for polyharmonic operators, integral equation, nonexistence of solutions
Mathematical Subject Classification 2010
Primary: 35J60
Secondary: 45G15
Milestones
Received: 9 September 2010
Revised: 30 May 2011
Accepted: 11 July 2011
Published: 21 January 2012
Authors
Guozhen Lu
Department of Mathematics
Wayne State University
Detroit MI 48202
United States
http://www.math.wayne.edu/~gzlu
Jiuyi Zhu
Department of Mathematics
Wayne State University
Detroit MI 48202
United States