  We present new $L^\infty$ a priori estimates for weak solutions of a wide
class of subcritical $p$-laplacian equations in bounded domains. No hypotheses
on the sign of the solutions, neither of the non-linearities are required. This
method is based in elliptic regularity for the $p$-laplacian combined either
with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation
inequalities.
  Let us consider a quasilinear boundary value problem $ -\Delta_p u= f(x,u),$
in $\Omega,$ with Dirichlet boundary conditions, where $\Omega \subset
\mathbb{R}^N $, with $p<N,$ is a bounded smooth domain strictly convex, and $f$
is a subcritical Carath\'eodory non-linearity. We provide $L^\infty$ a priori
estimates for weak solutions, in terms of their $L^{p^*}$-norm, where $p^*=
\frac{Np}{N-p}\ $ is the critical Sobolev exponent.
  By a subcritical non-linearity we mean, for instance, $|f(x,s)|\le
|x|^{-\mu}\, \tilde{f}(s),$ where $\mu\in(0,p),$ and
$\tilde{f}(s)/|s|^{p_{\mu}^*-1}\to 0$ as $|s|\to \infty$, here
$p^*_{\mu}:=\frac{p(N-\mu)}{N-p}$ is the critical Sobolev-Hardy exponent. Our
non-linearities includes non-power non-linearities.
  In particular we prove that when
$f(x,s)=|x|^{-\mu}\,\frac{|s|^{p^*_{\mu}-2}s}{\big[\log(e+|s|)\big]^\alpha}\,,$
with $\mu\in[1,p),$ then, for any $\varepsilon>0$ there exists a constant
$C_\varepsilon>0$ such that for any solution $u\in H^1_0(\Omega)$, the
following holds $$ \Big[\log\big(e+\|u\|_{\infty}\big)\Big]^\alpha\le
C_\varepsilon \, \Big(1+\|u\|_{p^*}\Big)^{\, (p^*_{\mu}-p)(1+\varepsilon)}\, ,
$$ where $C_\varepsilon$ is independent of the solution $u$.
