$L^{\infty }$ a-priori estimates for subcritical $p$-laplacian equations with a Carathéodory 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 $p<N,$ 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 $L^{p^{\ast }}$-norm, where $p^{\ast }=\frac{Np}{N-p}$ 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 $\tilde{f}(s)/|s{|}^{p_{\mu }^{\ast }-1}\to 0$ as $|s|\to \infty$, here $p_{\mu }^{\ast }:=\frac{p(N-\mu )}{N-p}$ is the critical Sobolev-Hardy exponent. Our non-linearities includes non-power non-linearities. In particular we prove that when $f(x,s)=|x{|}^{-\mu }\frac{|s{|}^{p_{\mu }^{\ast }-2}s}{[\log (e+|s|){]}^{\alpha }},$ with $\mu \in [1,p),$ then, for any $\epsilon >0$ there exists a constant $C_{\epsilon }>0$ such that for any solution $u\in H_0^1(\Omega )$, the following holds $$\left[\log \right(e+\Vert u{\Vert }_{\infty }){]}^{\alpha }\le C_{\epsilon }(1+\Vert u{\Vert }_{p^{\ast }}{)}^{(p_{\mu }^{\ast }-p)(1+\epsilon )},$$ where $C_{\epsilon }$ is independent of the solution $u$.

arXiv admin note: substantial text overlap with arXiv:2209.00073