What syntatic role does the noun 'desire' occupy in this infinitive clause?

Question

I know that infinitive clauses can be used as modifiers. Most of the time, I can easily identify their place. See this example:

[1] He found a place to sleep.

Although it isn't explicitly stated, my conclusion is that a preposition has been omitted for conciseness, that being 'in'. I can surmise this by changing the clause slightly:

[2] He found a place in which to sleep.

[3] He found a place to sleep in.

However, the following example confuses me:

[4] He had the desire to succeed.

Where does 'desire' fit in this clause? Could it be 'the desire with which to succeed'? It may be this simple, but it doesn't feel right. This also raises another question: does the noun phrase being modified by an infinitive clause need to play a part in the clause, or can the clause simply provide explanation (like a content/that- clause)?

To support this theory, relative clauses don't always feature the noun phrase itself. This is evident in the next example:

[5] He didn't understand the reason why she betrayed him.

The modified noun phrase, 'the reason', is not the antecedent of 'why'. This is a relative adverb, providing the answer in the form of an adverbial, such as the subordinating clause 'because [Insert Appropriate Explanation]'.

EDIT: John Lawler has informed me that 'the reason' is in fact the antecedent of 'why'. This was an ill-conceived presumption on my part.

Answer 1

Short answer provided by Araucaria:

The to-infinitival is not a relative clause there. It's the Complement of the noun desire.

The noun 'desire' does not have a syntactic function in the infinitive clause, as said clause isn't relative. Instead, the clause provides information that elaborates on the desire felt by the subject.

This is similar to a content clause (that- clause), which can function as a noun complement. For example, 'He hated the fact that everything wasn't going to plan.'

Answer 2

[1] He found a place to sleep. -> [1] He found a place for sleeping: The infinitive and gerund combined with a preposition give a modifier -> He found a sleeping place.

He had the desire to succeed. -> He had the desire for success. The infinitive and noun combined with a preposition give a modifier. We now know the sort of desire he had. However, the adjunct “to succeed/for success” cannot be translated into an adjective – it thus remains a complement that gives definition to "desire".