  We study Gibbs measures with log-correlated base Gaussian fields on the
$d$-dimensional torus. In the defocusing case, the construction of such Gibbs
measures follows from Nelson's argument. In this paper, we consider the
focusing case with a quartic interaction. Using the variational formulation, we
prove non-normalizability of the Gibbs measure. When $d = 2$, our argument
provides an alternative proof of the non-normalizability result for the
focusing $\Phi^4_2$-measure by Brydges and Slade (1996). We also go over the
construction of the focusing Gibbs measure with a cubic interaction. In the
appendices, we present (a) non-normalizability of the Gibbs measure for the
two-dimensional Zakharov system and (b) the construction of focusing quartic
Gibbs measures with smoother base Gaussian measures, showing a critical nature
of the log-correlated Gibbs measure with a focusing quartic interaction.
