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