Actually, this is not so much a case of missing esse, but of praedicative use of an adjectival word. Adjectives (solus), but also participles, can be used such that they agree with a nominal group (Iulius), while telling you something about the praedicate as a whole (Iulius non [habitat]), not just about the nominal group. This is also possible in other Indo-European languages:
She arrived late/crying/alone.
In English, it is not so easy to see whether a word is an adjective or an adverb, but the construction used is the same as in English. The subject she was alone, but she also arrived in a manner characterised as being alone. So alone tells you something about her, but also about her manner of arriving.
Subject and object complements are very common (trivial) cases of praedicative adjectives, in Latin, in English, and in many other languages:
Iulius est solus.
The lone Iulius is": this is clearly wrong and not how the sentence was intended.
"Iulius is alone": this is what it means.
So the adjective solus is not used attributively, i.e. it is not used to only modify the nominal group Iulius like any standard use of an adjective. Instead, it tells us something about how Iulius is; it tells you something about the praedicate Iulius est as a whole. Like any praedicative adjective, it still also tells you something about Iulius himself, in addition to modifying the praedicate.
In English, we often indicate praedicative use by placing the adjectival word after the nominal group it modifies, as in my first example, She arrived...