Let $\Omega \subset R^n$ a bounded domain. Consider $u \in L^1(\Omega)$. The definition of essential supremum is well known.
I am reading the book "Linear and quasilinear elliptic equations" of Ladyzhenskaya and Ural'tseva. The author use the concept of "essential maximum" and they don't define this concept. I am searching if this concept is the essential supremum but I am not finding anything. Someone know the definition and if is the same thing?
It's an unconventional choice of terminology, either by the authors or by the translators of the book (probably the former). Read "maximum" as "supremum" and "minimum" as "infimum", throughout the book.
This is not limited to "essential" versions; they also write $\max_\Omega u $ with no expectation of the maximum being attained.