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$L^\infty$ a-priori estimates for subcritical $p$-laplacian equations with a Carathéodory nonlinearity

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 $ -Δ_p u= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $p<N,$ is a bounded smooth domain strictly convex, and $f$ is a subcritical Carathéodory 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, $|f(x,s)|\le |x|^{-μ}\, \tilde{f}(s),$ where $μ\in(0,p),$ and $\tilde{f}(s)/|s|^{p_μ^*-1}\to 0$ as $|s|\to \infty$, here $p^*_μ:=\frac{p(N-μ)}{N-p}$ is the critical Sobolev-Hardy exponent. Our non-linearities includes non-power non-linearities. In particular we prove that when $f(x,s)=|x|^{-μ}\,\frac{|s|^{p^*_μ-2}s}{\big[\log(e+|s|)\big]^α}\,,$ with $μ\in[1,p),$ then, for any $\varepsilon>0$ there exists a constant $C_\varepsilon>0$ such that for any solution $u\in H^1_0(Ω)$, the following holds $$ \Big[\log\big(e+\|u\|_{\infty}\big)\Big]^α\le C_\varepsilon \, \Big(1+\|u\|_{p^*}\Big)^{\, (p^*_μ-p)(1+\varepsilon)}\, , $$ where $C_\varepsilon$ is independent of the solution $u$.