= Solution
The equation $F_\mu(x,y)=0$, together with $c(x)>0$ and $y>0$, is the perturbed <Karush-Kuhn-Tucker conditions>: stationarity is unchanged, while exact <complementary slackness> becomes $c_i(x)y_i=\mu$ for each $i$. A finite limit of such points as $\mu\downarrow0$ satisfies the original conditions by continuity.
For a strictly feasible $x$, the second block uniquely specifies $y_i=\mu/c_i(x)$. Substitution into the first block gives precisely $\nabla B_\mu(x)=0$. Thus
$$
\boxed{F_\mu(x,y)=0\iff x\text{ is a critical point of }B_\mu\text{ and }y_i=\mu/c_i(x),}
$$
within the stated strictly positive domain. These are stationary barrier points; they need not all be local minimizers in a nonconvex problem.
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