Convergence of generalized Orlicz norms with lower growth rate tending to infinity

We study convergence of generalized Orlicz energies when the lower growth-rate tends to infinity. We generalize results by Bocea–Mihăilescu (Orlicz case) and Eleuteri–Prinari (variable exponent case) and allow weaker assumptions: we are also able to handle unbounded domains with irregular boundary and non-doubling energies.