Approximation of \(L^\infty \) functionals with generalized Orlicz norms

The aim of this paper is to deal with the asymptotics of generalized Orlicz norms when the lower growth rate tends to infinity. We generalize results proven by Bertazzoni, Harjulehto and Hästö in Journ. of Math. Anal. and Appl. (2024) for integral type energies (in generalized Orlicz spaces), considering milder convexity assumptions. \(\Gamma \) $\mathrm{\Gamma }$ -convergence results and related representation theorems in terms of \(L^\infty \) $L^{\infty }$ functionals are proven. The convexity hypotheses are completely removed in the variable exponent setting, thus extending the results in Eleuteri-Prinari in Nonlinear Anal. Real. World Appl. (2021) and Prinari-Zappale in JOTA (2020).