An excellent post by the Phoenician Maverick on the difficulty of translating 'something exists' into Frege-Russell logic, as defended by Londonista David Brightly.
I don't agree with David 's approach, which is to invent the predicate 'Object(x)', a predicate which is satisfied by everything, and which fails to be satisfied by nothing. This translates 'something exists' into 'for some x, object(x)'. Neat - in Frege-Russell 'for all x F(x)' is true even when nothing exists. But unsatisfying, because there remains the difficulty of negative existential singular sentences. How do we translate 'Pegasus does not exist'? If 'Pegasus' is a name at all, then Object(Pegasus), which translates to 'Pegasus exists', which is false.
I prefer a non-standard logic which (as David knows) extends the Frege approach from general existential statements to singular statements. For Frege, 'serpents exist' means that the concept 'serpent' is instantiated. If we accept singular concepts, it follows that 'Pegasus exists' means that the concept Pegasus is instantiated, and 'something exists' means that some concept is instantiated.
For more on singular concepts, use this search key. Maverick has objected to the notion of singular concepts many times, but has failed to address any of my masterful replies.