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

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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

Abstract
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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.
[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.
[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.
[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.
[5]	M. F. Bidaut-Véron and P. Grillot, Singularities in elliptic systems with absorption terms,, \emph{Annali Scuola Norm. Sup. Pisa}, 28 (1999), 229.
[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.
[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).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	J. López-Gómez, Coexistence and metacoexistence for competitive species,, \emph{Houston J. Math.}, 29 (2003), 483.
[22]	V. Rădulescu, Singular phenomena in nonlinear elliptic problems,, in \emph{Handbook of differential equations; stationary partial differential equations}, (2007).
[23]	L. Véron, Semilinear elliptic equations with uniform blow up on the boundary,, \emph{J. Anal. Math.}, 59 (1992), 231. doi: 10.1007/BF02790229.

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.
[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.
[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.
[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.
[5]	M. F. Bidaut-Véron and P. Grillot, Singularities in elliptic systems with absorption terms,, \emph{Annali Scuola Norm. Sup. Pisa}, 28 (1999), 229.
[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.
[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).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	J. López-Gómez, Coexistence and metacoexistence for competitive species,, \emph{Houston J. Math.}, 29 (2003), 483.
[22]	V. Rădulescu, Singular phenomena in nonlinear elliptic problems,, in \emph{Handbook of differential equations; stationary partial differential equations}, (2007).
[23]	L. Véron, Semilinear elliptic equations with uniform blow up on the boundary,, \emph{J. Anal. Math.}, 59 (1992), 231. doi: 10.1007/BF02790229.

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