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A sharp integrability condition on the right-hand side of the  p -Laplace system for all its solutions to be continuous is exhibited. Their uniform continuity is also analyzed and estimates for their modulus of continuity are provided. The relevant estimates are shown to be optimal as the right-hand side ranges in classes of rearrangement-invariant spaces, such as Lebesgue, Lorentz, Lorentz–Zygmund, and Marcinkiewicz spaces, as well as some customary Orlicz spaces.

Continuity properties of solutions to the p -Laplace system