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

Abstract

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 $p 0$ 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 $u$.