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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

Marie-Françoise Bidaut-Véron 1, , Marta García-Huidobro 2, and  Cecilia Yarur 3,

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.
Keywords: Quasilinear elliptic systems, a priori estimates, asymptotic behaviour, Harnack inequality., large solutions.
Mathematics Subject Classification: Primary: 35B40, 35B45, 35J47; Secondary: 35J92, 35M3.
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.

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), 293-324. 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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