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Axionic holographic RG flow solutions are studied in the context of generalEinstein-Axion-Dilaton theories. A non-trivial axion profile is dual to the(non-perturbative) running of the $\theta$-term for the corresponding instantondensity operator. It is shown that a non-trivial axion solution is incompatiblewith a non-trivial (holographic) IR conformal fixed point. Imposing a suitableaxion regularity condition allows to select the IR geometry in a unique way.The solutions are found analytically in the asymptotic UV and IR regimes, andit is shown that in those regimes the axion backreaction is always negligible.The axion backreaction may become important in the intermediate region of thebulk. To make contact with the axion probe limit solutions, a systematicexpansion of the solution is developed. Several concrete examples are workedout numerically. It is shown that the regularity condition always implies afinite allowed range for the axion source parameter in the UV. This translatesinto the existence of a finite (but large) number of saddle-points in the large$N_c$ limit. This ties in well with axion-swampland conjectures.

Axion RG flows and the holographic dynamics of instanton densities