For the sentence 'I have been having difficulty sleeping', how would you analyse the verb phrase elements : VP have been having? 'have is an auxiliary - required to create -en form of the verb 'been'. but then the present progressive (having) also rquires an auxiliary [a form of to be]. Does this mean that 'been' is both a main verb, and an auxiliary? Seems weird!
Would you agree that "I have been having difficulty sleeping" is similar in form to "I have been reading"? By switching the verb to something more concrete, it becomes a little easier to parse.
"I have been reading" is present perfect continuous. Both "have" and "been" are auxiliary verbs. "Have" is the perfect auxiliary, and "been" is the progressive auxiliary. "Reading" is the main verb in present participle (-ing) form.
This is the extent of analysis found on most grammar websites. It seems like a good start for an English language learner, although it's likely possible to delve further into an advanced analysis of the role of auxiliary verbs within sentence structure. (There's also English Language & Usage)