  The classical Seiberg-Witten equation in dimesion three and four can be
generalized to an unifying framework, the generalized Seiberg-Witten (GSW)
equation. It includes the anti-self dulaity (ASD) equation, the $\mathrm
U(n)$-monopole equation, the Seiberg-Witten equation with multiple spinors, the
Vafa-Witten equation, the complex ASD equation. In this article, we prove some
vanishing results for solutions (called GSW monopoles and GSW Bogomolny
monopoles, respectively) of these equations with finite Yang-Mills-Higgs energy
over $\mathbf R^4$ and $\mathbf R^3$. We also prove a result on the growth of
the spinor in a GSW monopole over $\mathbf R^3$. These vanishing results
generalize the results obtained by Nagy and Oliveira [NO19] for Kapustin-Witten
equations.
