  As in the work of Tartar ( Tartar L. Interpolation non lin\'eaire et
r\'egularit\'e, 9, Journal of Functional Analysis, (1972), 469-489) we
developed here some new results on non linear interpolation of
$\alpha$-H\"olderian mappings between normed spaces, namely, by studying the
action of the mappings on $K$-functionals and between interpolation spaces with
logarithm functors. We apply those results to obtain regularity results on the
gradient of the solution to quasilinear equations of the form $$-div(\widehat
a(\nabla u ))+V(u)=f, $$ whenever $V$ is a nonlinear potential, $f$ belongs to
non standard spaces as Lorentz-Zygmund spaces. We show among other that the
mapping $T: \ Tf=\nabla u$ is locally or globally $\alpha$-H\"olderian under
suitable values of $\alpha$ and adequate hypothesis on $V$ and $\widehat a.$
