A numerically based investigation on the symmetry breaking and asymptotic behavior of the ground states to the p-Hénon equation
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The symmetry breaking phenomenon (SBP) to the Hénon equation was first numerically observed in  and then theoretically verified on the unit ball Bn in . Some results on the asymptotic behavior of the ground states to the Hénon equation on Bn are presented in [2, 3, 7].  further discussed SBP to the p-Hénon equation and obtained some results with special value p ≤ n on Bn. To inspire theoretical study on more general p, a series of numerical experiments to the p-Hénon 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. © 2011 Texas State University - San Marcos.
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