We study properties of Lipschitz truncations of Sobolev functions
with constant and variable exponent.
As non-trivial applications we use the
Lipschitz truncations to provide a simplified proof of an existence result for incompressible power-law like fluids presented in
[Frehse et al., SIAM J. Math. Anal
34 (2003) 1064–1083]. We also establish new existence results
to a class of incompressible electro-rheological fluids.