  Given a weakly compact cardinal $\kappa$, we give an axiomatization of
intuitionistic first-order logic over $\mathcal{L}_{\kappa^+, \kappa}$ and
prove it is sound and complete with respect to Kripke models. As a consequence
we get the disjunction and existence properties for that logic. This
generalizes the work of Nadel for intuitionistic logic over
$\mathcal{L}_{\omega_1, \omega}$. When $\kappa$ is a regular cardinal such that
$\kappa^{<\kappa}=\kappa$, we deduce, by an easy modification of the proof, a
complete axiomatization of intuitionistic first-order logic over
$\mathcal{L}_{\kappa^+, \kappa, \kappa}$, the language with disjunctions of at
most $\kappa$ formulas, conjunctions of less than $\kappa$ formulas and
quantification on less than $\kappa$ many variables. In particular, this
applies to any regular cardinal under the Generalized Continuum Hypothesis.
