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July  2013, 12(4): 1547-1568. doi: 10.3934/cpaa.2013.12.1547

## Keller-Osserman estimates for some quasilinear elliptic systems

 1 Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Faculté des Sciences, 37200 Tours 2 Departamento de Matemáticas, Pontiﬁcia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago 3 Departamento de Matemática y C.C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago

Received  February 2011 Revised  March 2012 Published  November 2012

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.
Citation: Marie-Françoise Bidaut-Véron, Marta García-Huidobro, Cecilia Yarur. Keller-Osserman estimates for some quasilinear elliptic systems. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1547-1568. doi: 10.3934/cpaa.2013.12.1547
##### References:
 [1] M-F. Bidaut-Véron, Local and global behaviour of solutions of quasilinear equations of Emden-Fowler type, Arch. Rat. Mech. Anal., 107 (1989), 293-324. doi: 10.1007/BF00251552. [2] M-F. Bidaut-Véron, Singularities of solutions of a class of quasilinear equations in divergence form, Nonlinear diffusion equations and their equilibrium states, Birkauser, Boston, Basel, Berlin, (1992), 129-144. [3] M-F. Bidaut-Véron, Removable singularities and existence for a quasilinear equation, Adv. Nonlinear Studies, 3 (2003), 25-63. [4] M-F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Diff. Eq., 15 (2010), 1033-1082. [5] M-F. Bidaut-Véron and P. Grillot, Singularities in elliptic systems with absorption terms, Ann. Scuola Norm. Sup. Pisa CL. Sci, 28 (1999), 229-271. [6] M-F. Bidaut-Véron and P. Grillot, Asymptotic behaviour of elliptic systems with mixed absorption and source terms, Asymtotic Anal., 19 (1999), 117-147. [7] M-F. Bidaut-Véron, M. Garcia-Huidobro and C. Yarur, Large solutions of elliptic systems of second order and applications to the biharmonic equation, Discrete and Continuous Dynamical Systems, 32 (2012), 411-432. [8] M-F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Mathématique, 84 (2001), 1-49. doi: 10.1007/BF02788105. [9] M-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539. doi: 10.1007/BF01243922. [10] L. d'Ambrosio and E. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic equations, Advances in Math., 224 (2010), 967-1020. doi: 10.1016/j.aim.2009.12.017. [11] J. Dávila, L. Dupaigne, O. Goubet and S. Martinez, Boundary blow-up solutions of cooperative systems, Ann. I. H. Poincaré-AN, 26 (2009), 1767-1791. [12] E. Di Benedetto, "Partial Differential Equations," Birkaüser, 1995. [13] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa, 28 (1999), 741-808. [14] A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations, J. Diff. Equ., 250 (2011), 4367-4408 and 4408-4436. [15] J. García-Melián, R. Letelier-Albornoz and J. Sabina de Lis, The solvability of an elliptic system under a singular boundary condition, Proc. Roy. Soc. Edinburgh, 136 (2006), 509-546. [16] J. García-Melian and J. Rossi, Boundary blow-up solutions to elliptic system of competitive type, J. Diff. Equ., 206 (2004), 156-181. doi: 10.1016/j.jde.2003.12.004. [17] J. García-Melián, Large solutions for an elliptic system of quasilinear equations, J. Diff. Equ., 245 (2008), 3735-3752. doi: 10.1016/j.jde.2008.04.004. [18] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure and Applied Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406. [19] J. B. Keller, On the solutions of $-\Delta u=f(u),$ Comm. Pure Applied Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402. [20] T. Kilpelainen and J. Maly, Degenerate elliptic equations with measure data and non linear potentials, Ann. Scuola Norm. Sup. Pisa, 19 (1992), 591-613. [21] T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Matematica, 172 (1994), 137-161. doi: 10.1007/BF02392793. [22] T. Kilpelainen and X. Zhong, Growth of entire $\mathcal{A}$-subharmonic functions, Ann. Acad. Sci. Fennic Math., 28 (2003), 181-192. [23] E. Mitidieri and S. Pohozaev, Absence of positive solutions for quasilinear elliptic problems on $\mathbb{R}^N2$, Proc. Steklov Institute of Math., 227 (1999), 186-216. [24] R. Osserman, On the inequality $-\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647. [25] J. Serrin, Local behavior of solutions of quasilinear equations, Acta Mathematica, 111 (1964), 247-302. doi: 10.1007/BF02391014. [26] J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Mathematica, 113 (1965), 219-240. doi: 10.1007/BF02391778. [27] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Mathematica, 189 (2002), 79-142. doi: 10.1007/BF02392645. [28] N. Trudinger, On Harnack type inequalities and their application to quasilinear equations, Comm. Pure Applied Math., 20 (1967), 721-747. doi: 10.1002/cpa.3160200406. [29] J. L. Vazquez, An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion, Nonlinear Anal., 5 (1981), 95-103. doi: 10.1016/0362-546X(81)90074-2. [30] J. L. Vazquez and L. Véron, Removable singularities of some strongly nonlinear elliptic equations, Manuscripta Math., 33 (1980), 129-144. doi: 10.1007/BF01316972. [31] L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., 59 (1992), 231-250. doi: 10.1007/BF02790229. [32] M. Wu and Z. Yang, Existence of boundary blow-up solutions for a class of quasilinear elliptic systems with critical case, Applied Math. Comput., 198 (2008), 574-581. doi: 10.1016/j.amc.2007.08.074.

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##### References:
 [1] M-F. Bidaut-Véron, Local and global behaviour of solutions of quasilinear equations of Emden-Fowler type, Arch. Rat. Mech. Anal., 107 (1989), 293-324. doi: 10.1007/BF00251552. [2] M-F. Bidaut-Véron, Singularities of solutions of a class of quasilinear equations in divergence form, Nonlinear diffusion equations and their equilibrium states, Birkauser, Boston, Basel, Berlin, (1992), 129-144. [3] M-F. Bidaut-Véron, Removable singularities and existence for a quasilinear equation, Adv. Nonlinear Studies, 3 (2003), 25-63. [4] M-F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Diff. Eq., 15 (2010), 1033-1082. [5] M-F. Bidaut-Véron and P. Grillot, Singularities in elliptic systems with absorption terms, Ann. Scuola Norm. Sup. Pisa CL. Sci, 28 (1999), 229-271. [6] M-F. Bidaut-Véron and P. Grillot, Asymptotic behaviour of elliptic systems with mixed absorption and source terms, Asymtotic Anal., 19 (1999), 117-147. [7] M-F. Bidaut-Véron, M. Garcia-Huidobro and C. Yarur, Large solutions of elliptic systems of second order and applications to the biharmonic equation, Discrete and Continuous Dynamical Systems, 32 (2012), 411-432. [8] M-F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Mathématique, 84 (2001), 1-49. doi: 10.1007/BF02788105. [9] M-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539. doi: 10.1007/BF01243922. [10] L. d'Ambrosio and E. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic equations, Advances in Math., 224 (2010), 967-1020. doi: 10.1016/j.aim.2009.12.017. [11] J. Dávila, L. Dupaigne, O. Goubet and S. Martinez, Boundary blow-up solutions of cooperative systems, Ann. I. H. Poincaré-AN, 26 (2009), 1767-1791. [12] E. Di Benedetto, "Partial Differential Equations," Birkaüser, 1995. [13] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa, 28 (1999), 741-808. [14] A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations, J. Diff. Equ., 250 (2011), 4367-4408 and 4408-4436. [15] J. García-Melián, R. Letelier-Albornoz and J. Sabina de Lis, The solvability of an elliptic system under a singular boundary condition, Proc. Roy. Soc. Edinburgh, 136 (2006), 509-546. [16] J. García-Melian and J. Rossi, Boundary blow-up solutions to elliptic system of competitive type, J. Diff. Equ., 206 (2004), 156-181. doi: 10.1016/j.jde.2003.12.004. [17] J. García-Melián, Large solutions for an elliptic system of quasilinear equations, J. Diff. Equ., 245 (2008), 3735-3752. doi: 10.1016/j.jde.2008.04.004. [18] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure and Applied Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406. [19] J. B. Keller, On the solutions of $-\Delta u=f(u),$ Comm. Pure Applied Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402. [20] T. Kilpelainen and J. Maly, Degenerate elliptic equations with measure data and non linear potentials, Ann. Scuola Norm. Sup. Pisa, 19 (1992), 591-613. [21] T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Matematica, 172 (1994), 137-161. doi: 10.1007/BF02392793. [22] T. Kilpelainen and X. Zhong, Growth of entire $\mathcal{A}$-subharmonic functions, Ann. Acad. Sci. Fennic Math., 28 (2003), 181-192. [23] E. Mitidieri and S. Pohozaev, Absence of positive solutions for quasilinear elliptic problems on $\mathbb{R}^N2$, Proc. Steklov Institute of Math., 227 (1999), 186-216. [24] R. Osserman, On the inequality $-\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647. [25] J. Serrin, Local behavior of solutions of quasilinear equations, Acta Mathematica, 111 (1964), 247-302. doi: 10.1007/BF02391014. [26] J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Mathematica, 113 (1965), 219-240. doi: 10.1007/BF02391778. [27] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Mathematica, 189 (2002), 79-142. doi: 10.1007/BF02392645. [28] N. Trudinger, On Harnack type inequalities and their application to quasilinear equations, Comm. Pure Applied Math., 20 (1967), 721-747. doi: 10.1002/cpa.3160200406. [29] J. L. Vazquez, An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion, Nonlinear Anal., 5 (1981), 95-103. doi: 10.1016/0362-546X(81)90074-2. [30] J. L. Vazquez and L. Véron, Removable singularities of some strongly nonlinear elliptic equations, Manuscripta Math., 33 (1980), 129-144. doi: 10.1007/BF01316972. [31] L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., 59 (1992), 231-250. doi: 10.1007/BF02790229. [32] M. Wu and Z. Yang, Existence of boundary blow-up solutions for a class of quasilinear elliptic systems with critical case, Applied Math. Comput., 198 (2008), 574-581. doi: 10.1016/j.amc.2007.08.074.
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