Approximation of $L^{\mathrm{\infty }}$ 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. $\mathrm{\Gamma }$-convergence results and related representation theorems in terms of $L^{\mathrm{\infty }}$ functionals are proven for sequences of generalized Orlicz energies under mild convexity assumptions. This latter hypothesis is removed in the variable exponent setting.