Local Boundedness of Minimizers with Limit Growth Conditions

Abstract

The energy integral of the calculus of variations, which we consider in this paper, has a limit behavior when the maximum exponent $$q$$q, in the growth estimate of the integrand, reaches a threshold. In fact, if $$q$$q is larger than this threshold, counterexamples to the local boundedness and regularity of minimizers are known. In this paper, we prove the local boundedness of minimizers (and also of quasi-minimizers) under this stated limit condition. Some other general and limit growth conditions are also considered.