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Second-order boundary estimates for large positive solutions to an elliptic system of competitive type. (English) Zbl 1479.35341

Summary: In this paper, we study the second-order boundary asymptotic behaviour for large positive solutions to an elliptic system of competitive type. First, we derive a second-order estimate to a related single weighted equation with boundary blow-up data. Then, by relaxing the system and iterating the estimate of the single equation, we establish second-order estimates of the solutions.

MSC:

35J47 Second-order elliptic systems
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35B40 Asymptotic behavior of solutions to PDEs
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[1] C. Anedda and G. Porru, Second order estimates for boundary blow-up solutions of elliptic equations, Discrete Contin. Dyn. Syst. 2007, Dynamical Systems and Differential Equations, Proceedings of the 6th AIMS International Conference, Suppl. 54-63. · Zbl 1163.35350
[2] C. Anedda and G. Porru, Boundary estimates for solutions of weighted semilinear elliptic equations, Discrete Contin. Dyn. Syst. A 32 (2012), no. 11, 3801-3817. · Zbl 1250.35104
[3] C. Bandle and M. Marcus, On second-order effects in the boundary behaviour of large solutions of semilinear elliptic problems, Differential Integral Equations 11 (1998), 23-34. · Zbl 1042.35535
[4] M. Chuaqui, C. Cortazar, M. Elgueta and J. Garcia-Melian, Uniqueness and boundary behaviour of large solutions to elliptic problems with singular weights, Comm. Pure Appl. Anal. 3 (2004), 653-662. · Zbl 1174.35386
[5] M. Chuaqui, C. Cortazar, M. Elgueta, C. Flores, J. Garcia-Melian and R. Letelier, On an elliptic problem with boundary blow-up and a singular weight: The radial case, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 6, 1283-1297. · Zbl 1039.35036
[6] M. Del Pino and R. Letelier, The influence of domain geometry in boundary blow-up elliptic problems, Nonlinear Anal. 48 (2002), 897-904. · Zbl 1142.35431
[7] J. Garcia-Melian, A remark on the existence of large solutions via sub and supersolutions, Electron. J. Differ. Equ. 110 (2003), 495-500. · Zbl 1040.35026
[8] J. Garcia-Melian, Uniqueness for boundary blow-up problems with continuous weights, Proc. Amer. Math. Soc. 135 (2007), 2785-2793. · Zbl 1146.35036
[9] E. Giarrusso and G. Porru, Second order estimates for boundary blow-up solutions of elliptic equations with an additive gradient term, Nonlinear Anal. 129 (2015), 160-172. · Zbl 1328.35059
[10] J. Garcia-Melian and J.D. Rossi, Boundary blow-up solutions to elliptic systems of competitive type, J. Differential Equations 206 (2004), 156-181. · Zbl 1162.35359
[11] S. Huang, Asymptotic behavior of boundary blow-up solutions to elliptic equations, Z. Angew. Math. Phys. 67 (2016), no. 1, 3 pp. · Zbl 1339.35114
[12] S. Huang, Q. Tian, S. Zhang, et al, A second-order estimate for blow-up solutions of elliptic equations, Nonlinear Anal. 74 (2011), 2342-2350. · Zbl 1210.35108
[13] B. Khamessi, Existence and asymptotic behavior of positive blow up solutions for semilinear elliptic coupled system, J. Math. Phys. 60 (2019), no. 9, 091505. · Zbl 1422.35060
[14] B. Khamessi and S.B. Othman, Exact boundary behavior of positive large solutions of a nonlinear Dirichlet problem, Nonlinear Anal. 187 (2019), 307-319. · Zbl 1427.35083
[15] A.V. Lair, A.W. Wood, Large solutions of semilinear elliptic problems, Nonlinear Anal. 37 (1999), 805-812. · Zbl 0932.35081
[16] C. Mu, S. Huang, Q. Tian, et al., Large solutions for an elliptic system of competitive type: Existence, uniqueness and asymptotic behavior, Nonlinear Anal. 71 (2009), 4544-4552. · Zbl 1167.35373
[17] L. Wei and Z. Yang, Large solutions of quasilinear elliptic system of competitive type: existence and asymptotic behavior, Int. J. Differ. Equ. 1 (2010), 1-17. · Zbl 1207.35125
[18] Z. Zhang, The second expansion of large solutions for semilinear elliptic equations, Nonlinear Anal. 74 (2011), 3445-3457. · Zbl 1217.35074
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