Perfect Substitution with a Continuum of goods?
Seems like the problem specified as specified right now should have as solution $C(\omega)=\frac{\delta(\omega)}{P(\omega)}$ if $P(\omega)=\min_{\omega'} P(\omega')$ and zero otherwise. The solution proposed cannot be optimal because the budget constraint is not depleted. If we let $\omega^*$ be the index of the cheapest good, the indirect utility would be then $U^*(P(w))=1/P(\omega^*)$.