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Keller-Osserman estimates for some quasilinear elliptic systems

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  • 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 $1 < p$, $ q < N$, and $D = \delta \mu- (p-1) (q-1) > 0$; the model case is $\mathcal{A}_p = \Delta_p$, $\mathcal{A}_q = \Delta_q.$ Despite of the lack of comparison principle, we prove a priori estimates of Keller-Osserman type: \begin{eqnarray} u(x)\leq Cd(x,\partial\Omega)^{-\frac{p(q-1)+q\delta}{D}},\qquad v(x)\leq Cd(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 the behaviour near 0 of the solutions of more general weighted systems, giving a priori estimates and removability results. Finally we prove the sharpness of the results.
    Mathematics Subject Classification: Primary: 35B40, 35B45, 35J47; Secondary: 35J92, 35M30.

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