= Soft-mode shell integral
In three dimensions, let $J(q)=r+\gamma(q^2-q_*^2)^2$, with $q_*>0$ and small $r>0$. Near the soft shell, $J(q)\simeq r+4\gamma q_*^2(q-q_*)^2$, so
$$
\int\frac{d^3q}{(2\pi)^3J(q)}
\sim\frac{q_*}{4\pi\sqrt{\gamma r}},
\qquad
\int\frac{q^2\,d^3q}{(2\pi)^3J(q)}
\sim\frac{q_*^3}{4\pi\sqrt{\gamma r}}.
$$
The radial integral uses $\int_{-\infty}^{\infty}ds/(r+As^2)=\pi/\sqrt{Ar}$. The $r^{-1/2}$ divergence obstructs the zero-mass limit of a self-consistent <Gaussian variational approximation> at a fixed nonzero ordering <wavevector>. This is the mechanism behind the <Brazovskii fluctuation-induced first-order transition>; https://arxiv.org/abs/cond-mat/0003026[Gross, Ignatiev and Chakraborty's study of fluctuation-driven ordering] analyzes the corresponding standard <Brazovskii model>.
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