In this paper, first, we will investigate the Dirichlet problem for one type of vortex equation, which generalizes the well-known Hermitian Einstein equation. Secondly, we will give existence results for solutions of these vortex equations over various complete noncompact K\"ahler manifolds.

Keywords:

vortex equation, Hermitian Yang--Mills--Higgs metric, holomorphic vector bundle, Kähler manifolds vortex equation, Hermitian Yang--Mills--Higgs metric, holomorphic vector bundle, Kähler manifolds

MSC Classifications:

58E15, 53C07 show english descriptions Application to extremal problems in several variables; Yang-Mills functionals [See also 81T13], etc.
Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) [See also 32Q20] 58E15 - Application to extremal problems in several variables; Yang-Mills functionals [See also 81T13], etc.
53C07 - Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) [See also 32Q20]

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