Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 20, pp. 1-23.
Title: A numerically based investigation on the symmetry breaking and
asymptotic behavior of the ground states to the p-Henon equation
Authors: Xudong Yao (Shanghai Normal Univ., Shanghai, China)
Jianxin Zhou (Texas A&M Univ., College Station, TX, USA
Abstract:
The symmetry breaking phenomenon (SBP) to the Henon
equation was first numerically observed in [4] and then
theoretically verified on the unit ball $B_n$ in [8]. Some
results on the asymptotic behavior of the ground states to the
Henon equation on $B_n$ are presented in [2,3,7].
[8] further discussed SBP to
the p-Henon equation and obtained some results with
special value $p\leq n$ on $B_n$. To inspire theoretical study on
more general p, a series of numerical experiments to the
p-Henon 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 1-peak solution, which is symmetric about
two axes and two diagonal lines, breaks its peak from 1 to 4, and a
1-peak positive non-ground 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 07, 2011.
Math Subject Classifications: 58E05, 58E30, 35A40, 35J65.
Key Words: The p-Henon equation; ground state; symmetry breaking;
peak breaking; asymptotic behavior.