Can we use letter with a subscript as a variable in Mathematica?

Yes you can, with limitations.

You have at least three different ways to make an assignment to a subscripted symbol a₀ :

1. make a rule for Subscript

2. make a rule for a

3. "symbolize" a₀ using the Notation package/palette

In each case below, when I write e.g. Subscript[a, 1] this can also be entered as a₁ by typing a then Ctrl+_ then 1.

When you write:

Subscript[a, 1] = "dog";

You make an assignment to Subscript:

DownValues[Subscript]

{HoldPattern[a₁] :> "dog"}

You make a rule for a by using TagSet:

a /: Subscript[a, 2] = "cat";

UpValues[a]

{HoldPattern[a₂] :> "cat"}

If you use the Notation palette you mess with underlying Box forms behind the scenes, allowing for assignment to OwnValues:

Each of these can be cleared with either Unset or TagUnset:

Subscript[a, 1] =.

a /: Subscript[a, 2] =.

You can also do this:

<< Notation`
Symbolize[ParsedBoxWrapper[SubscriptBox["_", "_"]]]

If you want to, you can import the Notation package first, then use the Symbolize function, so you don't have to use the ParsedBoxWrapper function, and just enter _,(ctrl+_),_. What this does, is set a pattern matching a subscripted character as a symbol. Mathematica will then treat expressions matching this form as a single variable.

This doesn't seem to affect functions that use subscripts in their definition such as the partial differential function.