The H\'enon equation, a generalized form of the Emden equation, admits symmetry-breaking bifurcation for a certain ratio of the transverse velocity to the radial velocity. Therefore, it has asymmetric solutions on a symmetric domain even though the Emden equation has no asymmetric unidirectional solution on such a domain. We discuss a numerical verification method for proving the existence of solutions of the H\'enon equation on a bounded domain. By applying the method to a line-segment domain and a square domain, we numerically prove the existence of solutions of the H\'enon equation for several parameters representing the ratio of transverse to radial velocity. As a result, we find a set of undiscovered solutions with three peaks on the square domain.
翻译:H\'enon 等式是一种通用的 Emden 等式形式, 允许对称破碎的对称分解对射速相对于射线速度的某种比例。 因此, 它在对称域上有不对称的解决方案, 尽管 Emden 等式在这样的域上没有不对称的单向解决方案。 我们讨论一个数字核查方法, 以证明在受约束域上存在 H\'enon 等式的解决方案。 通过对线分隔域和平方域应用该方法, 我们用数字来证明H'enon 等式在代表反向速度相对于射线速度的若干参数上存在解决方案。 结果, 我们找到了一组未发现的解决方案, 方域上有三个峰值 。