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March  2016, 15(2): 549-562. doi: 10.3934/cpaa.2016.15.549

Elliptic systems with boundary blow-up: existence, uniqueness and applications to removability of singularities

Jorge García-Melián 1, , Julio D. Rossi 2, and  José C. Sabina de Lis 3,

1. 

Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271. La Laguna
2. 

Departamento de Análisis Matemático, Universidad de Alicante, Ap 99, 03080, Alicante
3. 

Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 -- La Laguna, Spain

Received  July 2015 Revised  December 2015 Published  January 2016

In this paper we consider the elliptic system $\Delta u = u^p -v^q$, $\Delta v= -u^r +v^s$ in $\Omega$, where the exponents verify $p,s>1$, $q,r>0$ and $ps>qr$, and $\Omega$ is a smooth bounded domain of $R^N$. First, we show existence and uniqueness of boundary blow-up solutions, that is, solutions $(u,v)$ verifying $u=v=+\infty$ on $\partial \Omega$. Then, we use them to analyze the removability of singularities of positive solutions of the system in the particular case $qr\leq 1$, where comparison is available.
Keywords: boundary blow-up, removable singularities., Elliptic systems.
Mathematics Subject Classification: Primary: 35J47, 35J57; Secondary: 35B4.
Citation: Jorge García-Melián, Julio D. Rossi, José C. Sabina de Lis. Elliptic systems with boundary blow-up: existence, uniqueness and applications to removability of singularities. Communications on Pure & Applied Analysis, 2016, 15 (2) : 549-562. doi: 10.3934/cpaa.2016.15.549
References:
[1]

C. Bandle and M. Marcus, "Large" solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour,, \emph{J. Anal. Math.}, 58 (1992), 9.  doi: 10.1007/BF02790355.  Google Scholar

[2]

M. F. Bidaut-Véron, M. García-Huidobro and C. Yarur, Keller-Osserman estimates for some quasilinear elliptic systems,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 1547.   Google Scholar

[3]

M. F. Bidaut-Véron and P. Grillot, Estimations a priori pour les singularitées isolées d'un système elliptique hamiltonien,, \emph{C. R. Acad. Sci. Paris S\'er. I Math.}, 325 (1997), 617.  doi: 10.1016/S0764-4442(97)84771-4.  Google Scholar

[4]

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

[5]

M. F. Bidaut-Véron and P. Grillot, Singularities in elliptic systems with absorption terms,, \emph{Annali Scuola Norm. Sup. Pisa}, 28 (1999), 229.   Google Scholar

[6]

H. Brezis and L. Véron, Removable singularities for some nonlinear elliptic equations,, \emph{Arch. Rat. Mech. Anal.}, 75 (1980), 1.  doi: 10.1007/BF00284616.  Google Scholar

[7]

F. C. Cîrstea, A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials,, \emph{Mem. Amer. Math. Soc.}, 227 (2014).   Google Scholar

[8]

N. Dancer and Y. Du, Effects of certain degeneracies in the predator-prey model,, \emph{SIAM J. Math. Anal.}, 34 (2002), 292.  doi: 10.1137/S0036141001387598.  Google Scholar

[9]

J. Dávila, L. Dupaigne, O. Goubet and S. Martínez, Boundary blow-up solutions of cooperative systems,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 26 (2009), 1767.  doi: 10.1016/j.anihpc.2008.12.003.  Google Scholar

[10]

J. I. Díaz, M. Lazzo and P. G. Schmidt, Large solutions for a system of elliptic equations arising from fluid dynamics,, \emph{SIAM J. Math. Anal.}, 37 (2005), 490.  doi: 10.1137/S0036141004443555.  Google Scholar

[11]

G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness,, \emph{Nonl. Anal.}, 20 (1993), 97.  doi: 10.1016/0362-546X(93)90012-H.  Google Scholar

[12]

Y. Du, Effects of a degeneracy in the competition model. Part I: classical and generalized steady-state solutions,, \emph{J. Diff. Eqns.}, 181 (2002), 92.  doi: 10.1006/jdeq.2001.4074.  Google Scholar

[13]

Y. Du, Effects of a degeneracy in the competition model. Part II: perturbation and dynamical behaviour,, \emph{J. Diff. Eqns.}, 181 (2002), 133.  doi: 10.1006/jdeq.2001.4075.  Google Scholar

[14]

M. García-Huidobro and C. Yarur, Classification of positive singular solutions for a class of semilinear elliptic systems,, \emph{Adv. Diff. Eqns.}, 2 (1997), 383.   Google Scholar

[15]

J. García-Melián, A remark on uniqueness of large solutions for elliptic systems of competitive type,, \emph{J. Math. Anal. Appl.}, 331 (2007), 608.  doi: 10.1016/j.jmaa.2006.09.006.  Google Scholar

[16]

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

[17]

J. García-Melián, R. Letelier Albornoz and J. Sabina de Lis, The solvability of an elliptic system under a singular boundary condition,, \emph{Proc. Roy. Soc. Edinburgh}, 136 (2006), 509.  doi: 10.1017/S0308210500005047.  Google Scholar

[18]

J. García-Melián and J. D. Rossi, Boundary blow-up solutions to elliptic systems of competitive type,, \emph{J. Diff. Eqns.}, 206 (2004), 156.  doi: 10.1016/j.jde.2003.12.004.  Google Scholar

[19]

J. García-Melián and A. Suárez, Existence and uniqueness of positive large solutions to some cooperative elliptic systems,, \emph{Adv. Nonl. Stud.}, 3 (2003), 193.   Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[21]

J. López-Gómez, Coexistence and metacoexistence for competitive species,, \emph{Houston J. Math.}, 29 (2003), 483.   Google Scholar

[22]

V. Rădulescu, Singular phenomena in nonlinear elliptic problems,, in \emph{Handbook of differential equations; stationary partial differential equations}, (2007).   Google Scholar

[23]

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

show all references

References:
[1]

C. Bandle and M. Marcus, "Large" solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour,, \emph{J. Anal. Math.}, 58 (1992), 9.  doi: 10.1007/BF02790355.  Google Scholar

[2]

M. F. Bidaut-Véron, M. García-Huidobro and C. Yarur, Keller-Osserman estimates for some quasilinear elliptic systems,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 1547.   Google Scholar

[3]

M. F. Bidaut-Véron and P. Grillot, Estimations a priori pour les singularitées isolées d'un système elliptique hamiltonien,, \emph{C. R. Acad. Sci. Paris S\'er. I Math.}, 325 (1997), 617.  doi: 10.1016/S0764-4442(97)84771-4.  Google Scholar

[4]

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

[5]

M. F. Bidaut-Véron and P. Grillot, Singularities in elliptic systems with absorption terms,, \emph{Annali Scuola Norm. Sup. Pisa}, 28 (1999), 229.   Google Scholar

[6]

H. Brezis and L. Véron, Removable singularities for some nonlinear elliptic equations,, \emph{Arch. Rat. Mech. Anal.}, 75 (1980), 1.  doi: 10.1007/BF00284616.  Google Scholar

[7]

F. C. Cîrstea, A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials,, \emph{Mem. Amer. Math. Soc.}, 227 (2014).   Google Scholar

[8]

N. Dancer and Y. Du, Effects of certain degeneracies in the predator-prey model,, \emph{SIAM J. Math. Anal.}, 34 (2002), 292.  doi: 10.1137/S0036141001387598.  Google Scholar

[9]

J. Dávila, L. Dupaigne, O. Goubet and S. Martínez, Boundary blow-up solutions of cooperative systems,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 26 (2009), 1767.  doi: 10.1016/j.anihpc.2008.12.003.  Google Scholar

[10]

J. I. Díaz, M. Lazzo and P. G. Schmidt, Large solutions for a system of elliptic equations arising from fluid dynamics,, \emph{SIAM J. Math. Anal.}, 37 (2005), 490.  doi: 10.1137/S0036141004443555.  Google Scholar

[11]

G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness,, \emph{Nonl. Anal.}, 20 (1993), 97.  doi: 10.1016/0362-546X(93)90012-H.  Google Scholar

[12]

Y. Du, Effects of a degeneracy in the competition model. Part I: classical and generalized steady-state solutions,, \emph{J. Diff. Eqns.}, 181 (2002), 92.  doi: 10.1006/jdeq.2001.4074.  Google Scholar

[13]

Y. Du, Effects of a degeneracy in the competition model. Part II: perturbation and dynamical behaviour,, \emph{J. Diff. Eqns.}, 181 (2002), 133.  doi: 10.1006/jdeq.2001.4075.  Google Scholar

[14]

M. García-Huidobro and C. Yarur, Classification of positive singular solutions for a class of semilinear elliptic systems,, \emph{Adv. Diff. Eqns.}, 2 (1997), 383.   Google Scholar

[15]

J. García-Melián, A remark on uniqueness of large solutions for elliptic systems of competitive type,, \emph{J. Math. Anal. Appl.}, 331 (2007), 608.  doi: 10.1016/j.jmaa.2006.09.006.  Google Scholar

[16]

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

[17]

J. García-Melián, R. Letelier Albornoz and J. Sabina de Lis, The solvability of an elliptic system under a singular boundary condition,, \emph{Proc. Roy. Soc. Edinburgh}, 136 (2006), 509.  doi: 10.1017/S0308210500005047.  Google Scholar

[18]

J. García-Melián and J. D. Rossi, Boundary blow-up solutions to elliptic systems of competitive type,, \emph{J. Diff. Eqns.}, 206 (2004), 156.  doi: 10.1016/j.jde.2003.12.004.  Google Scholar

[19]

J. García-Melián and A. Suárez, Existence and uniqueness of positive large solutions to some cooperative elliptic systems,, \emph{Adv. Nonl. Stud.}, 3 (2003), 193.   Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[21]

J. López-Gómez, Coexistence and metacoexistence for competitive species,, \emph{Houston J. Math.}, 29 (2003), 483.   Google Scholar

[22]

V. Rădulescu, Singular phenomena in nonlinear elliptic problems,, in \emph{Handbook of differential equations; stationary partial differential equations}, (2007).   Google Scholar

[23]

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

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