What is the lightest excited state of the strongly selfcoupled Higgs field?

We argue for the existence of an upper bound $m_*$ on the Higgs mass at which the lowest excited state of the Higgs field ceases to be the conventional plane wave. An explicit construction of an alternative nonperturbative state is discussed. This excitation is spatially localized. The field fluctuations inside the localization region are large. The energy of the excitation is smaller than the mass of the plane wave state $m$ at $m>m_*$. An approximate value of $m_*$ is found to be $m_* \approx 4.75$ times the vacuum expectation value. This is an upper bound at the tree level.Comment: 11 pages, Dortmund University Do-TH-94/19, LaTex, 7 figures appended as uuencoded, tar-compressed .eps file