Geodesic
A numerically based investigation on the symmetry breaking and asymptotic behavior of the ground states to the $p$-Hénon equation
Electronic Journal of Differential Equations, Tome 2011 (2011).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 58E05, 58E30, 35A40, 35J65
Keywords: the p-henon equation, ground state, symmetry breaking, peak breaking, asymptotic behavior 
@article{EJDE_2011__2011__a39,
     author = {Yao, Xudong and Zhou, Jianxin},
     title = {A numerically based investigation on the symmetry breaking and asymptotic behavior of the ground states to the $p${-H\'enon} equation},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2011},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a39/}
}
TY  - JOUR
AU  - Yao, Xudong
AU  - Zhou, Jianxin
TI  - A numerically based investigation on the symmetry breaking and asymptotic behavior of the ground states to the $p$-Hénon equation
JO  - Electronic Journal of Differential Equations
PY  - 2011
VL  - 2011
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a39/
LA  - en
ID  - EJDE_2011__2011__a39
ER  - 
%0 Journal Article
%A Yao, Xudong
%A Zhou, Jianxin
%T A numerically based investigation on the symmetry breaking and asymptotic behavior of the ground states to the $p$-Hénon equation
%J Electronic Journal of Differential Equations
%D 2011
%V 2011
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a39/
%G en
%F EJDE_2011__2011__a39
Yao, Xudong; Zhou, Jianxin. A numerically based investigation on the symmetry breaking and asymptotic behavior of the ground states to the $p$-Hénon equation. Electronic Journal of Differential Equations, Tome 2011 (2011). http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a39/