Is local non-satiation enough to talk about Walrasian Equilibrium being subset of Core In Advanced Microeconomic Theory, the author mentions that the Cobb Douglas utility function is neither strongly increasing nor strictly quasiconcave over all of R+. But the condition of strongly increasing in R+ is used to prove that set of Walrasian Equilibrium allocations is a subset of Core. So we can't apply this result to Cobb Douglas utility functions to prove that Walrasian Equilibrium obtained is part of core. Is local non satiation sufficient to say set of Walrasian Equilibrium allocations is a subset of Core as Cobb Douglas satisfies local non satiation?
Yes, local-non satiation is sufficient for the result. In fact, the theorem holds as long as for every consumer every optimal consumption exhausts the budget. The wikipedia page has a proof which uses local non-satiation: https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics#Proof_of_the_first_theorem.

As a side note: Cobb-Douglas preferences do satisfy strict monotonicity on $\mathbb{R}^{n}_{++}$. Thus, if you can argue that you may restrict attention to bundles where all entries are strictly positive, the arguments based on strict monotonicity applies.