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We characterize, in the terms of intrinsic Hausdorff measures, the size of removable sets for Hölder continuous solutions to elliptic equations with Musielak-Orlicz growth. In the general case we provide a result in the new scale that is more relevant and captures -as special cases -the classical results, slightly refines the ones provided for problems stated in the variable exponent and double phase spaces and essentially improves the known one in the Orlicz case.2010 Mathematics Subject Classification. 35J60, 35J70.

We prove continuity up to the boundary of the minimizer of an obstacle problem and higher integrability of its gradient under generalized Orlicz growth. The result recovers similar results obtained in the special cases of polynomial growth, variable exponent growth and produces new results for Orlicz and double phase growth.

We prove local $C^{0,\alpha }$- and $C^{1,\alpha }$-regularity for the local solution to an obstacle problem with nonstandard growth. These results cover as special cases standard, variable exponent, double phase, and Orlicz growth.