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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 & 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.  doi: 10.1007/BF00251552.  Google Scholar

[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, (1992), 129.   Google Scholar

[3]

M-F. Bidaut-Véron, Removable singularities and existence for a quasilinear equation,, Adv. Nonlinear Studies, 3 (2003), 25.   Google Scholar

[4]

M-F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems,, Adv. Diff. Eq., 15 (2010), 1033.   Google Scholar

[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.   Google Scholar

[6]

M-F. Bidaut-Véron and P. Grillot, Asymptotic behaviour of elliptic systems with mixed absorption and source terms,, Asymtotic Anal., 19 (1999), 117.   Google Scholar

[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.   Google Scholar

[8]

M-F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems,, J. Anal. Math\'ematique, 84 (2001), 1.  doi: 10.1007/BF02788105.  Google Scholar

[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.  doi: 10.1007/BF01243922.  Google Scholar

[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.  doi: 10.1016/j.aim.2009.12.017.  Google Scholar

[11]

J. Dávila, L. Dupaigne, O. Goubet and S. Martinez, Boundary blow-up solutions of cooperative systems,, Ann. I. H. Poincar\'e-AN, 26 (2009), 1767.   Google Scholar

[12]

E. Di Benedetto, "Partial Differential Equations,", Birka\, (1995).   Google Scholar

[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.   Google Scholar

[14]

A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations,, J. Diff. Equ., 250 (2011), 4367.   Google Scholar

[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.   Google Scholar

[16]

J. García-Melian and J. Rossi, Boundary blow-up solutions to elliptic system of competitive type,, J. Diff. Equ., 206 (2004), 156.  doi: 10.1016/j.jde.2003.12.004.  Google Scholar

[17]

J. García-Melián, Large solutions for an elliptic system of quasilinear equations,, J. Diff. Equ., 245 (2008), 3735.  doi: 10.1016/j.jde.2008.04.004.  Google Scholar

[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.  doi: 10.1002/cpa.3160340406.  Google Scholar

[19]

J. B. Keller, On the solutions of $-\Delta u=f(u),$, Comm. Pure Applied Math., 10 (1957), 503.  doi: 10.1002/cpa.3160100402.  Google Scholar

[20]

T. Kilpelainen and J. Maly, Degenerate elliptic equations with measure data and non linear potentials,, Ann. Scuola Norm. Sup. Pisa, 19 (1992), 591.   Google Scholar

[21]

T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Matematica, 172 (1994), 137.  doi: 10.1007/BF02392793.  Google Scholar

[22]

T. Kilpelainen and X. Zhong, Growth of entire $\mathcalA$-subharmonic functions,, Ann. Acad. Sci. Fennic Math., 28 (2003), 181.   Google Scholar

[23]

E. Mitidieri and S. Pohozaev, Absence of positive solutions for quasilinear elliptic problems on $\mathbbR^N$,, Proc. Steklov Institute of Math., 227 (1999), 186.   Google Scholar

[24]

R. Osserman, On the inequality $-\Delta u\geq f(u)$,, Pacific J. Math., 7 (1957), 1641.   Google Scholar

[25]

J. Serrin, Local behavior of solutions of quasilinear equations,, Acta Mathematica, 111 (1964), 247.  doi: 10.1007/BF02391014.  Google Scholar

[26]

J. Serrin, Isolated singularities of solutions of quasilinear equations,, Acta Mathematica, 113 (1965), 219.  doi: 10.1007/BF02391778.  Google Scholar

[27]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Mathematica, 189 (2002), 79.  doi: 10.1007/BF02392645.  Google Scholar

[28]

N. Trudinger, On Harnack type inequalities and their application to quasilinear equations,, Comm. Pure Applied Math., 20 (1967), 721.  doi: 10.1002/cpa.3160200406.  Google Scholar

[29]

J. L. Vazquez, An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion,, Nonlinear Anal., 5 (1981), 95.  doi: 10.1016/0362-546X(81)90074-2.  Google Scholar

[30]

J. L. Vazquez and L. Véron, Removable singularities of some strongly nonlinear elliptic equations,, Manuscripta Math., 33 (1980), 129.  doi: 10.1007/BF01316972.  Google Scholar

[31]

L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary,, J. Anal. Math., 59 (1992), 231.  doi: 10.1007/BF02790229.  Google Scholar

[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.  doi: 10.1016/j.amc.2007.08.074.  Google Scholar

show all references

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.  doi: 10.1007/BF00251552.  Google Scholar

[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, (1992), 129.   Google Scholar

[3]

M-F. Bidaut-Véron, Removable singularities and existence for a quasilinear equation,, Adv. Nonlinear Studies, 3 (2003), 25.   Google Scholar

[4]

M-F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems,, Adv. Diff. Eq., 15 (2010), 1033.   Google Scholar

[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.   Google Scholar

[6]

M-F. Bidaut-Véron and P. Grillot, Asymptotic behaviour of elliptic systems with mixed absorption and source terms,, Asymtotic Anal., 19 (1999), 117.   Google Scholar

[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.   Google Scholar

[8]

M-F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems,, J. Anal. Math\'ematique, 84 (2001), 1.  doi: 10.1007/BF02788105.  Google Scholar

[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.  doi: 10.1007/BF01243922.  Google Scholar

[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.  doi: 10.1016/j.aim.2009.12.017.  Google Scholar

[11]

J. Dávila, L. Dupaigne, O. Goubet and S. Martinez, Boundary blow-up solutions of cooperative systems,, Ann. I. H. Poincar\'e-AN, 26 (2009), 1767.   Google Scholar

[12]

E. Di Benedetto, "Partial Differential Equations,", Birka\, (1995).   Google Scholar

[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.   Google Scholar

[14]

A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations,, J. Diff. Equ., 250 (2011), 4367.   Google Scholar

[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.   Google Scholar

[16]

J. García-Melian and J. Rossi, Boundary blow-up solutions to elliptic system of competitive type,, J. Diff. Equ., 206 (2004), 156.  doi: 10.1016/j.jde.2003.12.004.  Google Scholar

[17]

J. García-Melián, Large solutions for an elliptic system of quasilinear equations,, J. Diff. Equ., 245 (2008), 3735.  doi: 10.1016/j.jde.2008.04.004.  Google Scholar

[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.  doi: 10.1002/cpa.3160340406.  Google Scholar

[19]

J. B. Keller, On the solutions of $-\Delta u=f(u),$, Comm. Pure Applied Math., 10 (1957), 503.  doi: 10.1002/cpa.3160100402.  Google Scholar

[20]

T. Kilpelainen and J. Maly, Degenerate elliptic equations with measure data and non linear potentials,, Ann. Scuola Norm. Sup. Pisa, 19 (1992), 591.   Google Scholar

[21]

T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Matematica, 172 (1994), 137.  doi: 10.1007/BF02392793.  Google Scholar

[22]

T. Kilpelainen and X. Zhong, Growth of entire $\mathcalA$-subharmonic functions,, Ann. Acad. Sci. Fennic Math., 28 (2003), 181.   Google Scholar

[23]

E. Mitidieri and S. Pohozaev, Absence of positive solutions for quasilinear elliptic problems on $\mathbbR^N$,, Proc. Steklov Institute of Math., 227 (1999), 186.   Google Scholar

[24]

R. Osserman, On the inequality $-\Delta u\geq f(u)$,, Pacific J. Math., 7 (1957), 1641.   Google Scholar

[25]

J. Serrin, Local behavior of solutions of quasilinear equations,, Acta Mathematica, 111 (1964), 247.  doi: 10.1007/BF02391014.  Google Scholar

[26]

J. Serrin, Isolated singularities of solutions of quasilinear equations,, Acta Mathematica, 113 (1965), 219.  doi: 10.1007/BF02391778.  Google Scholar

[27]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Mathematica, 189 (2002), 79.  doi: 10.1007/BF02392645.  Google Scholar

[28]

N. Trudinger, On Harnack type inequalities and their application to quasilinear equations,, Comm. Pure Applied Math., 20 (1967), 721.  doi: 10.1002/cpa.3160200406.  Google Scholar

[29]

J. L. Vazquez, An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion,, Nonlinear Anal., 5 (1981), 95.  doi: 10.1016/0362-546X(81)90074-2.  Google Scholar

[30]

J. L. Vazquez and L. Véron, Removable singularities of some strongly nonlinear elliptic equations,, Manuscripta Math., 33 (1980), 129.  doi: 10.1007/BF01316972.  Google Scholar

[31]

L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary,, J. Anal. Math., 59 (1992), 231.  doi: 10.1007/BF02790229.  Google Scholar

[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.  doi: 10.1016/j.amc.2007.08.074.  Google Scholar

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