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Large solutions for an elliptic system of quasilinear equations. (English) Zbl 1169.35021
Summary: We consider the quasilinear elliptic system Δ p u=u a v b , Δ p v=u c v e in a smooth bounded domain Ω N , with the boundary conditions u=v=+ on Ω. The operator Δ p stands for the p-Laplacian defined by Δ p u=div(|u| p-2 u), p>1, and the exponents verify a,e>p-1, b,c>0 and (a-p+1)(e-p+1)bc. We analyze positive solutions in both components, providing necessary and sufficient conditions for existence. We also prove uniqueness of positive solutions in the case (a-p+1)(e-p+1)>bc and obtain the exact blow-up rate near the boundary of the solution. In the case (a-p+1)(e-p+1)=bc, infinitely many positive solutions are constructed.
MSC:
35J55Systems of elliptic equations, boundary value problems (MSC2000)
35J60Nonlinear elliptic equations
References:
[1]Bandle, C.; Marcus, M.: ’Large’ solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour, J. anal. Math. 58, 9-24 (1992) · Zbl 0802.35038 · doi:10.1007/BF02790355
[2]Bieberbach, L.: Δu=eu und die automorphen funktionen, Math. ann. 77, 173-212 (1916)
[3]Chuaqui, M.; Cortázar, C.; Elgueta, M.; Flores, C.; García-Melián, J.; Letelier, R.: On an elliptic problem with boundary blow-up and a singular weight: the radial case, Proc. roy. Soc. Edinburgh 133, 1283-1297 (2003) · Zbl 1039.35036 · doi:10.1017/S0308210500002936
[4]Chuaqui, M.; Cortázar, C.; Elgueta, M.; García-Melián, J.: Uniqueness and boundary behaviour of large solutions to elliptic problems with singular weights, Commun. pure appl. Anal. 3, 653-662 (2004) · Zbl 1174.35386 · doi:10.3934/cpaa.2004.3.653
[5]Cîrstea, F.; Du, Y.: General uniqueness results and variation speed for blow-up solutions of elliptic equations, Proc. London math. Soc. 91, 459-482 (2005) · Zbl 1108.35068 · doi:10.1112/S0024611505015273
[6]Cîrstea, F.; Rǎdulescu, V.: Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Acad. sci. Paris sér. I math. 335, No. 5, 447-452 (2002) · Zbl 1183.35124 · doi:10.1016/S1631-073X(02)02503-7
[7]Cîrstea, F.; Rǎdulescu, V.: Nonlinear problems with singular boundary conditions arising in population dynamics: A karamata regular variation theory approach, Asymptot. anal. 46, 275-298 (2006)
[8]Damascelli, L.: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. inst. H. Poincaré anal. Non linéaire 15, 493-516 (1998) · Zbl 0911.35009 · doi:10.1016/S0294-1449(98)80032-2 · doi:numdam:AIHPC_1998__15_4_493_0
[9]Dancer, N.; Du, Y.: Effects of certain degeneracies in the predator – prey model, SIAM J. Math. anal. 34, No. 2, 292-314 (2002) · Zbl 1055.35046 · doi:10.1137/S0036141001387598
[10]Del Pino, M.; Letelier, R.: The influence of domain geometry in boundary blow-up elliptic problems, Nonlinear anal. 48, No. 6, 897-904 (2002) · Zbl 1142.35431 · doi:10.1016/S0362-546X(00)00222-4
[11]Díaz, G.; Letelier, R.: Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonlinear anal. 20, 97-125 (1993) · Zbl 0793.35028 · doi:10.1016/0362-546X(93)90012-H
[12]Díaz, J. I.; Saa, E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. sci. Paris sér. I math. 305, 521-524 (1987) · Zbl 0656.35039
[13]Di Benedetto, E.: C1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear anal. 7, 827-850 (1983) · Zbl 0539.35027 · doi:10.1016/0362-546X(83)90061-5
[14]Du, Y.: Effects of a degeneracy in the competition model. Part I: Classical and generalized steady-state solutions, J. differential equations 181, 92-132 (2002) · Zbl 1042.35016 · doi:10.1006/jdeq.2001.4074
[15]Du, Y.: Effects of a degeneracy in the competition model. Part II: Perturbation and dynamical behaviour, J. differential equations 181, 133-164 (2002) · Zbl 1042.35017 · doi:10.1006/jdeq.2001.4075
[16]Du, Y.: Asymptotic behavior and uniqueness results for boundary blow-up solutions, Differential integral equations 17, 819-834 (2004) · Zbl 1150.35369
[17]Du, Y.; Guo, Z.: Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. anal. Math. 89, 277-302 (2003) · Zbl 1162.35028 · doi:10.1007/BF02893084
[18]García-Melián, J.: Nondegeneracy and uniqueness for boundary blow-up elliptic problems, J. differential equations 223, 208-227 (2006) · Zbl 1170.35405 · doi:10.1016/j.jde.2005.05.001
[19]García-Melián, J.: Uniqueness for boundary blow-up problems with continuous weights, Proc. amer. Math. soc. 135, 2785-2793 (2007) · Zbl 1146.35036 · doi:10.1090/S0002-9939-07-08822-3
[20]García-Melián, J.: A remark on uniqueness of large solutions for elliptic systems of competitive type, J. math. Anal. appl. 331, 608-616 (2007) · Zbl 1131.35015 · doi:10.1016/j.jmaa.2006.09.006
[21]J. García-Melián, Large solutions for equations involving the p-Laplacian and singular weights, submitted for publication · Zbl 1169.35332 · doi:10.1007/s00033-008-7141-z
[22]J. García-Melián, Quasilinear equations with boundary blow-up and exponential reaction, submitted for publication · Zbl 1177.35087
[23]García-Melián, J.; Letelier-Albornoz, R.; De Lis, J. Sabina: Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, Proc. amer. Math. soc. 129, No. 12, 3593-3602 (2001) · Zbl 0989.35044 · doi:10.1090/S0002-9939-01-06229-3
[24]García-Melián, J.; Letelier-Albornoz, R.; De Lis, J. Sabina: The solvability of an elliptic system under a singular boundary condition, Proc. roy. Soc. Edinburgh 136, 509-546 (2006)
[25]García-Melián, J.; Rossi, J. D.: Boundary blow-up solutions to elliptic systems of competitive type, J. differential equations 206, 156-181 (2004) · Zbl 1162.35359 · doi:10.1016/j.jde.2003.12.004
[26]J. García-Melián, J.D. Rossi, J. Sabina de Lis, Large solutions for the Laplacian with a power nonlinearity given by a variable exponent, Ann. Inst. H. Poincaré (C) Nonlinear Analysis, in press · Zbl 1177.35072 · doi:10.1016/j.anihpc.2008.03.007
[27]García-Melián, J.; Suárez, A.: Existence and uniqueness of positive large solutions to some cooperative elliptic systems, Adv. nonlinear stud. 3, 193-206 (2003) · Zbl 1045.35025
[28]Guo, Z.; Webb, J. R. L.: Structure of boundary blow-up solutions of quasilinear elliptic problems. I: large and small solutions, Proc. roy. Soc. Edinburgh sect. A 135, 615-642 (2005) · Zbl 1129.35381 · doi:10.1017/S0308210500004030 · doi:http://www.ingentaconnect.com/content/rse/proca/2005/00000135/00000003/art00009
[29]Guo, Z.; Webb, J. R. L.: Structure of boundary blow-up solutions of quasilinear elliptic problems, II: Small and intermediate solutions, J. differential equations 211, 187-217 (2005) · Zbl 1134.35339 · doi:10.1016/j.jde.2004.06.008
[30]Keller, J. B.: On solutions of Δu=f(u), Comm. pure appl. Math. 10, 503-510 (1957) · Zbl 0090.31801 · doi:10.1002/cpa.3160100402
[31]Kondrat’ev, V. A.; Nikishkin, V. A.: Asymptotics, near the boundary, of a solution of a singular boundary value problem for a semilinear elliptic equation, Differ. equ. 26, 345-348 (1990) · Zbl 0706.35054
[32]Ladyzhenskaya, O. A.; Ural’tseva, N. N.: Linear and quasilinear elliptic equations, (1968) · Zbl 0164.13002
[33]Li, H.; Wang, M.: Existence and uniqueness of positive solutions to the boundary blow-up problem for an elliptic system, J. differential equations 234, 246-266 (2007) · Zbl 1220.35039 · doi:10.1016/j.jde.2006.11.003
[34]Lieberman, G.: Boundary regularity for solutions of degenerate elliptic equations, Nonlinear anal. 12, 1203-1219 (1988) · Zbl 0675.35042 · doi:10.1016/0362-546X(88)90053-3
[35]Loewner, C.; Nirenberg, L.: Partial differential equations invariant under conformal of projective transformations, , 245-272 (1974) · Zbl 0298.35018
[36]López-Gómez, J.: The boundary blow-up rate of large solutions, J. differential equations 195, 25-45 (2003) · Zbl 1130.35329 · doi:10.1016/j.jde.2003.06.003
[37]López-Gómez, J.: Coexistence and metacoexistence for competitive species, Houston J. Math. 29, No. 2, 483-536 (2003) · Zbl 1034.35062
[38]López-Gómez, J.: Optimal uniqueness theorems and exact blow-up rates of large solutions, J. differential equations 224, 385-439 (2006) · Zbl 1208.35036 · doi:10.1016/j.jde.2005.08.008
[39]Marcus, M.; Véron, L.: Uniqueness and asymptotic behaviour of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. inst. H. Poincaré anal. Non linéaire 14, No. 2, 237-274 (1997) · Zbl 0877.35042 · doi:10.1016/S0294-1449(97)80146-1 · doi:numdam:AIHPC_1997__14_2_237_0
[40]Matero, J.: Quasilinear elliptic equations with boundary blow-up, J. anal. Math. 69, 229-247 (1996) · Zbl 0893.35032 · doi:10.1007/BF02787108
[41]Mckenna, P. J.; Reichel, W.; Walter, W.: Symmetry and multiplicity for nonlinear elliptic differential equations with boundary blow-up, Nonlinear anal. 28, 1213-1225 (1997) · Zbl 0868.35031 · doi:10.1016/S0362-546X(97)82870-2
[42]Mohammed, A.: Existence and asymptotic behavior of blow-up solutions to weighted quasilinear equations, J. math. Anal. appl. 298, 621-637 (2004) · Zbl 1126.35029 · doi:10.1016/j.jmaa.2004.05.030
[43]Mohammed, A.; Porcu, G.; Porru, G.: Large solutions to some non-linear ODE with singular coefficients, Nonlinear anal. 47, 513-524 (2001) · Zbl 1042.34534 · doi:10.1016/S0362-546X(01)00196-1
[44]Osserman, R.: On the inequality Δuf(u), Pacific J. Math. 7, 1641-1647 (1957) · Zbl 0083.09402
[45]Tolksdorf, P.: The Dirichlet problem in domains with conical boundary points, Comm. partial differential equations 8, 773-817 (1983)
[46]Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations, J. differential equations 51, 126-150 (1984) · Zbl 0488.35017 · doi:10.1016/0022-0396(84)90105-0
[47]Vázquez, J. L.: A strong maximum principle for some quasilinear elliptic equations, Appl. math. Optim. 12, 191-202 (1984) · Zbl 0561.35003 · doi:10.1007/BF01449041
[48]Véron, L.: Semilinear elliptic equations with uniform blowup on the boundary, J. anal. Math. 59, 231-250 (1992) · Zbl 0802.35042 · doi:10.1007/BF02790229
[49]Zhang, Z.: A remark on the existence of explosive solutions for a class of semilinear elliptic equations, Nonlinear anal. 41, 143-148 (2000) · Zbl 0964.35053 · doi:10.1016/S0362-546X(98)00270-3