In this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact K\"ahler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

MSC Classifications:

58J05, 53C07 show english descriptions Elliptic equations on manifolds, general theory [See also 35-XX]
Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) [See also 32Q20] 58J05 - Elliptic equations on manifolds, general theory [See also 35-XX]
53C07 - Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) [See also 32Q20]

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