# American Institute of Mathematical Sciences

February  2012, 32(2): 411-432. doi: 10.3934/dcds.2012.32.411

## Large solutions of elliptic systems of second order and applications to the biharmonic equation

 1 Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Faculté des Sciences, 37200 Tours, France 2 Departamento de Matemáticas, Pontiﬁcia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile 3 Departamento de Matemática y C.C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

Received  December 2010 Revised  May 2011 Published  September 2011

In this work we study the nonnegative solutions of the elliptic system $\Delta u=|x|^{a}v^{\delta},\qquad\Delta v=|x|^{b}u^{\mu}%$ in the superlinear case $\mu\delta>1,$ which blow up near the boundary of a domain of $\mathbb{R}^{N},$ or at one isolated point. In the radial case we give the precise behavior of the large solutions near the boundary in any dimension $N$. We also show the existence of infinitely many solutions blowing up at $0.$ Furthermore, we show that there exists a global positive solution in $\mathbb{R}^{N}\backslash\left\{ 0\right\} ,$ large at $0,$ and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation $\Delta^{2}u=\left\vert x\right\vert ^{b}\left\vert u\right\vert ^{\mu}.$ Our results are based on a new dynamical approach of the radial system by means of a quadratic system of order 4, introduced in [4], combined with the nonradial upper estimates of [5].
Citation: Marie-Françoise Bidaut-Véron, Marta García-Huidobro, Cecilia Yarur. Large solutions of elliptic systems of second order and applications to the biharmonic equation. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 411-432. doi: 10.3934/dcds.2012.32.411
##### References:
 [1] C. Bandle and M. Essén, On the solutions of quasilinear elliptic problems with boundary blow-up, in "Partial Differential Equations of Elliptic Type" (Cortona, 1992), Sympos. Math., XXXV, Cambrige Univ. Press, Cambridge, (1994), 93-111. [2] C. Bandle and M. Marcus, "Large" solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behavior, J. Anal. Math., 58 (1992), 9-24. doi: 10.1007/BF02790355. [3] C. Bandle and M. Marcus, On second-order effects in the boundary behavior of large solutions of semilinear elliptic problems, Differential and Integral Equations, 11 (1998), 23-34. [4] M. Bidaut-Veron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Diff. Eq., 15 (2010), 1033-1082. [5] M.-F. Bidaut-Veron and P. Grillot, Singularities in elliptic systems with absorption terms, Ann. Scuola Norm. Sup. Pisa CL. Sci., 28 (1999), 229-271. [6] W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations,'' D. C. Heath and Co., Boston, Mass., 1965. [7] O. Costin and L. Dupaigne, Boundary blow-up solutions in the unit ball: Asymptotics, uniqueness and symmetry, J. Diff. Equat., 249 (2010), 931-964. [8] J. Dávila, L. Dupaigne, O. Goubet and S. Martinez, Boundary blow-up solutions of cooperative systems, Ann. I. H. Poincaré Anal. Non Linéaire, 26 (2009), 1767-1791 [9] M. Del Pino and R. Letelier, The infuence of domain geometry in boundary blow-up elliptic problems, Nonlinear Anal., 48 (2002), 897-904. doi: 10.1016/S0362-546X(00)00222-4. [10] G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal., 20 (1993), 97-125. doi: 10.1016/0362-546X(93)90012-H. [11] J. I. Díaz, M. Lazzo and P. G. Schmidt, "Large Radial Solutions of a Plolyharmonic Equation with Superlinear Growth," Proceedings of the 2006 International Conference in honor of Jacqueline Feckinger, Electronic J. Diff. Equat., Conference 16 Texas State Univ.-San Marcos, Dept. Math., San Marcos, TX, (2007), 103-128. [12] J. García-Melián and A. Suárez, Existence and uniqueness of positive large solutions to some cooperative elliptic systems, Advanced Nonlinear Studies, 3 (2003), 193-206. [13] J. García-Melián and J. D. Rossi, Boundary blow-up solutions to elliptic systems of competitive type, J. Diff. Equat., 206 (2004), 156-181. [14] J. García-Melián, Large solutions for an elliptic system of quasilinear equations, J. Differential Equat., 245 (2008), 3735-3752. doi: 10.1016/j.jde.2008.04.004. [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 Sect. A, 136 (2006), 509-546. doi: 10.1017/S0308210500005047. [16] A. C. Lazer and P. J. Mckenna, Asymptotic behavior of solutions of boundary blowup problems, Differential and Integral Equations, 7 (1994), 1001-1019. [17] C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in "Contributions to Analysis" (A collection of papers dedicated to Lipman Bers), Acad. Press, New York, (1974), 245-272. [18] H. Logemann and E. P. Ryan, Non-autonomous systems: Asymptotic behavior and weak invariance principles, Journal of Diff. Equat., 189 (2003), 440-460. doi: 10.1016/S0022-0396(02)00144-4. [19] M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 237-274. [20] M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equations, J. Evol. Equ., 3 (2003), 637-652. doi: 10.1007/s00028-003-0122-y. [21] C. Mu, S. Huang, Q. Tian and L. Liu, Large solutions for an elliptic system of competitive type: Existence, uniqueness and asymptotic behavior, Nonlinear Anal., 71 (2009), 4544-4552. doi: 10.1016/j.na.2009.03.012. [22] L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., 59 (1992), 231-250. [23] L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations," Pitman Research Notes in Math. Series, 353, Longman, Harlow, 1996. [24] Y. Wang, Boundary blow-up solutions for a cooperative system of quasinear equation, preprint. [25] 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. [26] C. Yarur, Nonexistence of positive singular solutions for a class of semilinear elliptic systems, Electronic J. Differential Equat., 1996, approx. 22 pp (electronic). [27] C. Yarur, A priori estimates for positive solutions for a class of semilinear elliptic systems, Nonlinear Anal., 36 (1999), 71-90. doi: 10.1016/S0362-546X(97)00726-8.

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##### References:
 [1] C. Bandle and M. Essén, On the solutions of quasilinear elliptic problems with boundary blow-up, in "Partial Differential Equations of Elliptic Type" (Cortona, 1992), Sympos. Math., XXXV, Cambrige Univ. Press, Cambridge, (1994), 93-111. [2] C. Bandle and M. Marcus, "Large" solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behavior, J. Anal. Math., 58 (1992), 9-24. doi: 10.1007/BF02790355. [3] C. Bandle and M. Marcus, On second-order effects in the boundary behavior of large solutions of semilinear elliptic problems, Differential and Integral Equations, 11 (1998), 23-34. [4] M. Bidaut-Veron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Diff. Eq., 15 (2010), 1033-1082. [5] M.-F. Bidaut-Veron and P. Grillot, Singularities in elliptic systems with absorption terms, Ann. Scuola Norm. Sup. Pisa CL. Sci., 28 (1999), 229-271. [6] W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations,'' D. C. Heath and Co., Boston, Mass., 1965. [7] O. Costin and L. Dupaigne, Boundary blow-up solutions in the unit ball: Asymptotics, uniqueness and symmetry, J. Diff. Equat., 249 (2010), 931-964. [8] J. Dávila, L. Dupaigne, O. Goubet and S. Martinez, Boundary blow-up solutions of cooperative systems, Ann. I. H. Poincaré Anal. Non Linéaire, 26 (2009), 1767-1791 [9] M. Del Pino and R. Letelier, The infuence of domain geometry in boundary blow-up elliptic problems, Nonlinear Anal., 48 (2002), 897-904. doi: 10.1016/S0362-546X(00)00222-4. [10] G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal., 20 (1993), 97-125. doi: 10.1016/0362-546X(93)90012-H. [11] J. I. Díaz, M. Lazzo and P. G. Schmidt, "Large Radial Solutions of a Plolyharmonic Equation with Superlinear Growth," Proceedings of the 2006 International Conference in honor of Jacqueline Feckinger, Electronic J. Diff. Equat., Conference 16 Texas State Univ.-San Marcos, Dept. Math., San Marcos, TX, (2007), 103-128. [12] J. García-Melián and A. Suárez, Existence and uniqueness of positive large solutions to some cooperative elliptic systems, Advanced Nonlinear Studies, 3 (2003), 193-206. [13] J. García-Melián and J. D. Rossi, Boundary blow-up solutions to elliptic systems of competitive type, J. Diff. Equat., 206 (2004), 156-181. [14] J. García-Melián, Large solutions for an elliptic system of quasilinear equations, J. Differential Equat., 245 (2008), 3735-3752. doi: 10.1016/j.jde.2008.04.004. [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 Sect. A, 136 (2006), 509-546. doi: 10.1017/S0308210500005047. [16] A. C. Lazer and P. J. Mckenna, Asymptotic behavior of solutions of boundary blowup problems, Differential and Integral Equations, 7 (1994), 1001-1019. [17] C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in "Contributions to Analysis" (A collection of papers dedicated to Lipman Bers), Acad. Press, New York, (1974), 245-272. [18] H. Logemann and E. P. Ryan, Non-autonomous systems: Asymptotic behavior and weak invariance principles, Journal of Diff. Equat., 189 (2003), 440-460. doi: 10.1016/S0022-0396(02)00144-4. [19] M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 237-274. [20] M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equations, J. Evol. Equ., 3 (2003), 637-652. doi: 10.1007/s00028-003-0122-y. [21] C. Mu, S. Huang, Q. Tian and L. Liu, Large solutions for an elliptic system of competitive type: Existence, uniqueness and asymptotic behavior, Nonlinear Anal., 71 (2009), 4544-4552. doi: 10.1016/j.na.2009.03.012. [22] L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., 59 (1992), 231-250. [23] L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations," Pitman Research Notes in Math. Series, 353, Longman, Harlow, 1996. [24] Y. Wang, Boundary blow-up solutions for a cooperative system of quasinear equation, preprint. [25] 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. [26] C. Yarur, Nonexistence of positive singular solutions for a class of semilinear elliptic systems, Electronic J. Differential Equat., 1996, approx. 22 pp (electronic). [27] C. Yarur, A priori estimates for positive solutions for a class of semilinear elliptic systems, Nonlinear Anal., 36 (1999), 71-90. doi: 10.1016/S0362-546X(97)00726-8.
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