Keller-Osserman estimates for some quasilinear elliptic systems

Abstract

In this article we study quasilinear systems of two types, in a domain$\Omega$ of $R^N$ : with absorption terms, or mixed terms: \begin{eqnarray}(A): \mathcal{A}_{p} u=v^{\delta}, \mathcal{A}_{q}v=u^{\mu},\\(M): \mathcal{A}_{p} u=v^{\delta}, -\mathcal{A}_{q}v=u^{\mu},\end{eqnarray}where $\delta$, $\mu>0$ and $1 0$; the modelcase is $\mathcal{A}_p = \Delta_p$, $\mathcal{A}_q = \Delta_q.$ Despite ofthe lack of comparison principle, we prove a priori estimates ofKeller-Osserman type:\begin{eqnarray}u(x)\leq Cd(x,\partial\Omega)^{-\frac{p(q-1)+q\delta}{D}},\qquad v(x)\leqCd(x,\partial\Omega)^{-\frac{q(p-1)+p\mu}{D}}.\end{eqnarray}Concerning system $(M)$, we show that $v$ always satisfies Harnack inequality.In the case $\Omega=B(0,1)\backslash \{0\}$, we also study thebehaviour near 0 of the solutions of more general weighted systems, giving apriori estimates and removability results. Finally we prove the sharpness ofthe results.