\(L^\infty \) a-priori estimates for subcritical p-laplacian equations with a Carathéodory non-linearity

Let us consider a quasi-linear boundary value problem \( -\Delta _p u= f(x,u),\) $-{\mathrm{\Delta }}_{p}u=f(x,u),$ in \(\Omega ,\) $\mathrm{\Omega },$ with Dirichlet boundary conditions, where \(\Omega \subset \mathbb {R}^N \) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ , with \(p<N,\) $p<N,$ is a bounded smooth domain strictly convex, and the non-linearity f is a Carathéodory function p-super-linear and subcritical. We provide \(L^\infty \) $L^{\infty }$ a priori estimates for weak solutions, in terms of their \(L^{p^*}\) $L^{p^{\ast }}$ -norm, where \(p^*= \frac{Np}{N-p}\ \) $p^{\ast }=\frac{\mathit{Np}}{N-p}$ is the critical Sobolev exponent. 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 with Caffarelli–Kohn–Nirenberg interpolation inequalities. By a subcritical non-linearity we mean, for instance, \(|f(x,s)|\le |x|^{-\mu }\, \tilde{f}(s),\) $|f(x,s)|\le {|x|}^{-\mu }\tilde{f}(s),$ where \(\mu \in (0,p),\) $\mu \in (0,p),$ and \(\tilde{f}(s)/|s|^{p_{\mu }^*-1}\rightarrow 0\) $\tilde{f}(s)/{|s|}^{p_{\mu }^{\ast }-1}\to 0$ as \(|s|\rightarrow \infty \) $|s|\to \infty$ , here \(p^*_{\mu }:=\frac{p(N-\mu )}{N-p}\) $p_{\mu }^{\ast }:=\frac{p(N-\mu )}{N-p}$ is the critical Hardy–Sobolev exponent. Our non-linearities includes non-power non-linearities. In particular we prove that when \(f(x,s)=|x|^{-\mu }\,\frac{|s|^{p^*_{\mu }-2}s}{\big [\log (e+|s|)\big ]^\alpha },\) $f(x,s)={|x|}^{-\mu }\frac{{|s|}^{p_{\mu }^{\ast }-2}s}{[log(e+|s|){]}^{\alpha }},$ with \(\mu \in [1,p),\) $\mu \in [1,p),$ then, for any \(\varepsilon >0\) $\epsilon >0$ there exists a constant \(C_\varepsilon >0\) $C_{\epsilon }>0$ such that for any solution \(u\in H^1_0(\Omega )\) $u\in H_0^1(\mathrm{\Omega })$ , the following holds \(\begin{aligned} \Big [\log \big (e+\Vert u\Vert _{\infty }\big )\Big ]^\alpha \le C_\varepsilon \, \Big (1+\Vert u\Vert _{p^*}\Big )^{\, (p^*_{\mu }-p)(1+\varepsilon )}\,, \end{aligned}\) $$\begin{array}{c}\hfill [log(e+{\Vert u\Vert }_{\infty }){]}^{\alpha }\le C_{\epsilon }(1+{\Vert u\Vert }_{p^{\ast }}{)}^{(p_{\mu }^{\ast }-p)(1+\epsilon )},\end{array}$$ where \(C_\varepsilon \) $C_{\epsilon }$ is independent of the solution u.