A NUMERICALLY BASED INVESTIGATION ON THE SYMMETRY BREAKING AND ASYMPTOTIC BEHAVIOR OF THE GROUND STATES TO THE p-HENON EQUATION

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 <= 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.