# American Institute of Mathematical Sciences

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

 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.
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.

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##### 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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