It’s easy to pronounce the letter r in lernen. But when it comes to pronouncing lernst, it becomes really hard to keep pronouncing r next to nst.
Listening to some native speaker pronunciations (like this), I realized that r is omitted and not pronounced in lernst, which makes its pronunciation much easier. Is this only a special case where r is omitted, or is there any general pronunciation rule which is being applied here?
The /r/ is pronounced differently by different speakers. The three most common pronunciatons for /r/ in German today are (ordered by frequency of occurrence):
There are other, rare and locally limited pronunciations, among them a retroflex approximant [ɻ] similar to the American English pronunciation. These are listed in the Duden Grammatik.
What you must note (and this will answer your question) is that one speaker will pronounce all instances of /r/ in the same way. This means that if you listen to one individual speaker, the /r/ in both "lernen" and "lernst" will sound the same.
Which pronunciation you choose, is up to you and depends on the variant of German you want to speak.
My claim that "one speaker will pronounce all instances of /r/ in the same way" is obviously wrong. All native German speakers will realize this, if they observe themselves speaking the words (1) "breit", (2) "lernst" and (3) "Lager". There is a decrease of "roll" from (1) to (3). Maybe someone wants to dig up the rules to this distribution.
What you have here is a postvocalic r -- an r that follows a vowel. Like in British English, postvocalic r in German isn't pronounced "properly", but it does change the vowel it follows and is itself realized as a vowel (see also @Veri's answer).
Compare the following English word pairs for examples (and pronounce them British in your head):
bee -- beer
moo -- moor
The "ler" part of lernen and lernst rhymes with British English "fair".
Compare also what Wikipedia writes about the pronunciation of R in German: "Nicht überall dort, wo der Buchstabe R in der Schrift erscheint, wird er auch tatsächlich als das Phonem /r/ ausgesprochen. Meist findet sich in Wörterbüchern, die sich an die sogenannte Standardlautung des Deutschen orientieren, die Empfehlung zur Aussprache des Buchstabens als /r/ unter anderem nach den kurzen Vokalen i, ä, a, ü, ö, u, o am Wortende oder vor einem Konsonanten, z. B. [vɪr] für „wirr“ und nicht [vɪɐ̯].1 In anderen Fällen wird meist ein abgeschwächtes a [ɐ̯] angegeben, wie bei „Tür“, „wir“ oder „Mutter“."
I recorded an example how to pronounce it:
https://soundcloud.com/splattne/pronunciation-of-lernst
There are two slightly different ways:
I think it depends on whether you speak some special kind of dialect or not. I am from Austria, and we often pronounce the /r/-sound like an /a/-sound -> leanst. Unfortunately, I can't give you more information about this specific questions.
Man kann das r mehr oder weniger betonen oder rollen, aber auslassen kann man es nicht.
Auch in dem Audioschnipsel meine ich ein r zu hören.
