The maximum principle and positive principal eigen- functions for polyharmonic equations

For bounded smooth domains the number λ1 is the so-called principal eigenvalue. It has a unique eigenfunction, which is positive, and this eigenfunction is the only positive one (up to normalization). Two main references for this type of results, which are usually called maximum principles, are the books by Walter (1964) and by Protter and Weinberger (1967). Extensions to general bounded non-smooth domains are studied by Berestycki, Nirenberg and Varadhan (1994). Due to Clément and Peletier (1979) there is even a property which is called the anti-maximum principle.