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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 and 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, J. Anal. Math., 58 (1992), 9-24. doi: 10.1007/BF02790355.

[2]

M. F. Bidaut-Véron, M. García-Huidobro and C. Yarur, Keller-Osserman estimates for some quasilinear elliptic systems, Comm. Pure Appl. Anal., 12 (2013) (4), 1547-1568.

[3]

M. F. Bidaut-Véron and P. Grillot, Estimations a priori pour les singularitées isolées d'un système elliptique hamiltonien, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 617-622. 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, Asymp. Anal., 19 (1999), 117-147.

[5]

M. F. Bidaut-Véron and P. Grillot, Singularities in elliptic systems with absorption terms, Annali Scuola Norm. Sup. Pisa, 28 (1999), 229-271.

[6]

H. Brezis and L. Véron, Removable singularities for some nonlinear elliptic equations, Arch. Rat. Mech. Anal., 75 (1980), 1-6. doi: 10.1007/BF00284616.

[7]

F. C. Cîrstea, A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials, Mem. Amer. Math. Soc., 227 (2014), n. 1068.

[8]

N. Dancer and Y. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal., 34 (2002), 292-314. doi: 10.1137/S0036141001387598.

[9]

J. Dávila, L. Dupaigne, O. Goubet and S. Martínez, Boundary blow-up solutions of cooperative systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1767-1791. 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, SIAM J. Math. Anal., 37 (2005), 490-513. doi: 10.1137/S0036141004443555.

[11]

G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonl. Anal., 20 (1993), 97-125. 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, J. Diff. Eqns., 181 (2002), 92-132. doi: 10.1006/jdeq.2001.4074.

[13]

Y. Du, Effects of a degeneracy in the competition model. Part II: perturbation and dynamical behaviour, J. Diff. Eqns., 181 (2002), 133-164. 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, Adv. Diff. Eqns., 2 (1997), 383-402.

[15]

J. García-Melián, A remark on uniqueness of large solutions for elliptic systems of competitive type, J. Math. Anal. Appl., 331 (2007), 608-616. doi: 10.1016/j.jmaa.2006.09.006.

[16]

J. García-Melián, Large solutions for an elliptic system of quasilinear equations, J. Diff. Eqns., 245 (2008), 3735-3752. 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, Proc. Roy. Soc. Edinburgh, 136 (2006), 509-546. doi: 10.1017/S0308210500005047.

[18]

J. García-Melián and J. D. Rossi, Boundary blow-up solutions to elliptic systems of competitive type, J. Diff. Eqns., 206 (2004), 156-181. 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, Adv. Nonl. Stud., 3 (2003), 193-206.

[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, Houston J. Math., 29 (2003), 483-536.

[22]

V. Rădulescu, Singular phenomena in nonlinear elliptic problems, in Handbook of differential equations; stationary partial differential equations, vol. 4. (ed. M. Chipot), Elsevier, 2007.

[23]

L. Véron, Semilinear elliptic equations with uniform blow up on the boundary, J. Anal. Math., 59 (1992), 231-250. 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, J. Anal. Math., 58 (1992), 9-24. doi: 10.1007/BF02790355.

[2]

M. F. Bidaut-Véron, M. García-Huidobro and C. Yarur, Keller-Osserman estimates for some quasilinear elliptic systems, Comm. Pure Appl. Anal., 12 (2013) (4), 1547-1568.

[3]

M. F. Bidaut-Véron and P. Grillot, Estimations a priori pour les singularitées isolées d'un système elliptique hamiltonien, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 617-622. 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, Asymp. Anal., 19 (1999), 117-147.

[5]

M. F. Bidaut-Véron and P. Grillot, Singularities in elliptic systems with absorption terms, Annali Scuola Norm. Sup. Pisa, 28 (1999), 229-271.

[6]

H. Brezis and L. Véron, Removable singularities for some nonlinear elliptic equations, Arch. Rat. Mech. Anal., 75 (1980), 1-6. doi: 10.1007/BF00284616.

[7]

F. C. Cîrstea, A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials, Mem. Amer. Math. Soc., 227 (2014), n. 1068.

[8]

N. Dancer and Y. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal., 34 (2002), 292-314. doi: 10.1137/S0036141001387598.

[9]

J. Dávila, L. Dupaigne, O. Goubet and S. Martínez, Boundary blow-up solutions of cooperative systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1767-1791. 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, SIAM J. Math. Anal., 37 (2005), 490-513. doi: 10.1137/S0036141004443555.

[11]

G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonl. Anal., 20 (1993), 97-125. 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, J. Diff. Eqns., 181 (2002), 92-132. doi: 10.1006/jdeq.2001.4074.

[13]

Y. Du, Effects of a degeneracy in the competition model. Part II: perturbation and dynamical behaviour, J. Diff. Eqns., 181 (2002), 133-164. 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, Adv. Diff. Eqns., 2 (1997), 383-402.

[15]

J. García-Melián, A remark on uniqueness of large solutions for elliptic systems of competitive type, J. Math. Anal. Appl., 331 (2007), 608-616. doi: 10.1016/j.jmaa.2006.09.006.

[16]

J. García-Melián, Large solutions for an elliptic system of quasilinear equations, J. Diff. Eqns., 245 (2008), 3735-3752. 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, Proc. Roy. Soc. Edinburgh, 136 (2006), 509-546. doi: 10.1017/S0308210500005047.

[18]

J. García-Melián and J. D. Rossi, Boundary blow-up solutions to elliptic systems of competitive type, J. Diff. Eqns., 206 (2004), 156-181. 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, Adv. Nonl. Stud., 3 (2003), 193-206.

[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, Houston J. Math., 29 (2003), 483-536.

[22]

V. Rădulescu, Singular phenomena in nonlinear elliptic problems, in Handbook of differential equations; stationary partial differential equations, vol. 4. (ed. M. Chipot), Elsevier, 2007.

[23]

L. Véron, Semilinear elliptic equations with uniform blow up on the boundary, J. Anal. Math., 59 (1992), 231-250. doi: 10.1007/BF02790229.

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