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 。