Electron. J. Diff. Equ.,
Vol. 2011 (2011), No. 20, pp. 123.
A numerically based investigation on the symmetry breaking and
asymptotic behavior of the ground states to the pHenon equation
Xudong Yao, Jianxin Zhou
Abstract:
The symmetry breaking phenomenon (SBP) to the Henon
equation was first numerically observed in [4] and then
theoretically verified on the unit ball
in [8]. Some
results on the asymptotic behavior of the ground states to the
Henon equation on
are presented in [2,3,7].
[8] further discussed SBP to
the pHenon equation and obtained some results with
special value
on
.
To inspire theoretical study on
more general p, a series of numerical experiments to the
pHenon equation on a disk and a square are carried out
in this paper. Numerical computations are made by the minimax method
developed in [9,10]. Then, SBP, a peak break
phenomenon (PBP); i.e., a 1peak solution, which is symmetric about
two axes and two diagonal lines, breaks its peak from 1 to 4, and a
1peak positive nonground state solution, which is only symmetric
about one axis, on the square are numerically captured and
visualized. The peak point and the peak height of the ground states
are carefully calculated to study their asymptotic behavior. Several
conjectures are made based on the numerical observations to
stimulate theoretical analysis. Two of them are proved in this paper.
Submitted November 27, 2009. Published February 7, 2011.
Math Subject Classifications: 58E05, 58E30, 35A40, 35J65.
Key Words: The pHenon equation; ground state; symmetry breaking;
peak breaking; asymptotic behavior.
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Xudong Yao
Department of Mathematics,
Shanghai Normal University
Shanghai, 200234, China
email: xdyao@shnu.edu.cn

Jianxin Zhou
Department of Mathematics, Texas A&M University
College Station, TX 77843, USA
email: jzhou@math.tamu.edu

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