  We study the continuity of weak solutions for quasilinear elliptic systems
with source terms of critical growth arising from a transport-energy structure.
The latter occurs frequently in connection with the first balance principles of
physics for internal energy or entropy. Relying on the properties of diffusive
measures, we prove a H\"older continuity result for these PDE systems in the
case of vectorial p-growth and of low regularity features such as mixed
boundary conditions on Lipschitz domains. This result is valid in all space
dimensions. In the special case where the principal part is linear in the
second order, we also study the higher integrability of the gradients and the
uniqueness principle in the small. Finally two illustrations in the context of
electrothermal coupling are considered, the stationary thermistor model and the
Nernst-Planck equations of electrochemistry with variable temperature.
