Some Symmetry Results for the A-Laplacian Equation via the Moving Planes Method

Zhongbo Fang, Anna Wang

School of Mathematical Sciences, Ocean University of China, Qingdao, China.

DOI: 10.4236/apm.2012.26053                

Abstract

In this paper, we are concerned with a positive solution of the non-homogeneous A-Laplacian equation in an open bounded connected domain. We use moving planes method to prove that the domain is a ball and the solution is radially symmetric.

Keywords

Symmetry; A-Laplacian; Moving Planes Method; Overdetermined Boundary Value Problem

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. Serrin, “A Symmetry Problem in Potential Theory,” Archive for Rational Mechanics and Analysis, Vol. 43, No. 4, 1971, pp. 304-318. doi:10.1007/BF00250468
[2]	N. Garofalo and J. Lewis, “A Symmetry Result Related to Some Overdetermined Boundary Value Problems,” American Journal of Mathematics, Vol. 111, No. 1, 1989, pp. 9-33. doi:10.2307/2374477
[3]	I. Fragala, I. F. Gazzaola and B. Kawohl, “Overdetemined Boundary Value Problems with Possibly Degenerate Ellipticity: A Geometry Approach,” Mathematische Zeitschrift, Vol. 254, No. 1, 2006, pp. 117-132. doi:10.1007/s00209-006-0937-7
[4]	G. A. Philippin, “Application of the Maximum Principle to a Variety of Problems Involving Elliptic Differential Equations,” In: P. W. Schaefer, Ed., Maximum Principles and Eigenvalue Problems in Partial Differential Equations, Pitman Research Notes in Mathematics Series, Longman SciTech., Harlow, 1988, pp. 34-48.
[5]	A. Farina and B. Kawohl, “Remarks on an Overdetermined Boundary Value Problem,” Calculus of Variations and Partial Differential Equations, Vol. 31, No. 3, 2008, pp. 351-357.