Asymptotic behavior of fractional Musielak–Orlicz–Sobolev modulars without the \(\Delta _2\)-condition

In this article, we study the asymptotic behavior of anisotropic nonlocal nonstandard growth seminorms and modulars as the fractional parameter goes to 1 without requiring the \(\Delta _2\) ${\mathrm{\Delta }}_2$ -condition on the generalized Young function or its complementary function. This provides a so-called Bourgain-Brezis-Mironescu type formula for a very general family of functionals. In the particular case of fractional Sobolev spaces with variable exponent, we point out that our proof asks for a weaker regularity of the exponent than the considered in previous articles.