"A good application"
Translation:En god søknad
Søknad would be what you send to apply for a job or funds etc.
The phone application is too new to be in the dictionary but en app is the common term. En app, appen, apper, appene I think is the official one. (Although my hunch is that a lot of people writes it with an apostrophe because of the foreigness of the word en app, app'en, app'er, app'ene. Looks horrible in print though, the English word and an English apostrophe with the Norwegian endings added...)
If 'app' isn't in the dictionary, there can't be any official inflection. (Although it would probably be 'app-appen-apper-appene', as you say)
The use of apostrophes is not regarded as a standard by any dictionaries I know of, and should not be recommended.
As alek_d said, the correct word would be 'applikasjon', but 'app' is equally common nowadays, so it will probably be accepted as an alternative in a few years.