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November  2011, 10(6): 1733-1745. doi: 10.3934/cpaa.2011.10.1733

Blow-up rates of large solutions for semilinear elliptic equations

1. 

Department of Mathematics and Informational Science, Yantai University, P.O. Box 264005, Yantai, Shandong

2. 

School of Mathematical Science, Peking University, Beijing, 100871, China

Received  February 2010 Revised  April 2011 Published  May 2011

In this paper we analyze the blow-up rates of large solutions to the semilinear elliptic problem $\Delta u =b(x)f(u), x\in \Omega, u|_{\partial \Omega} = +\infty,$ where $\Omega$ is a bounded domain with smooth boundary in $R^N$, $f$ is rapidly varying or normalised regularly varying with index $p$ ($p>1$) at infinity, and $b \in C^\alpha (\bar{\Omega})$ which is non-negative in $\Omega$ and positive near the boundary and may be vanishing on the boundary.
Citation: Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733
References:
[1]

C. Anedda and G. Porru, Second order estimates for boundary blow-up solutions of elliptic equations, Discrete Contin. Dyn. Syst., (Suppl.) (2007), 54-63.

[2]

C. Anedda and G. Porru, Boundary behaviour for solutions of boundary blow-up problems in a borderline case, J. Math. Anal. Appl., 352 (2009), 35-47. doi: 10.1016/j.jmaa.2008.02.042.

[3]

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.

[4]

C. Bandle, Asymptotic behavior of large solutions of quasilinear elliptic problems, Z. angew. Math. Phys., 54 (2003), 731-738. doi: 10.1007/s00033-003-3207-0.

[5]

N. H. Bingham, C. M. Goldie and J. L. Teugels, "Regular Variation," Encyclopedia of Mathematics and its Applications 27, Cambridge University Press, Cambridge, 1987.

[6]

F. Cîrstea and V. D. Rădulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Acad. Sci. Paris, Sér. I, 335 (2002), 447-452. doi: 10.1112/S1631-073X(02)02523-7/FLA.

[7]

F. Cirstea and Y. Du, General uniqueness results and variation speed for blow-up solutions of elliptic equations, Proc. London Math. Soc., 91 (2005), 459-482. doi: 10.1112/S0024611505015273.

[8]

F. Cîrstea, Elliptic equations with competing rapidly varying nonlinearities and boundary blow-up, Advances in Differential Equations, 12 (2007), 995-1030.

[9]

H. Dong, S. Kim and M. Safonov, On uniqueness of boundary blow-up solutions of a class of nonlinear elliptic equations, Comm. Partial Diff. Equations, 33 (2008), 177-188. doi: 10.1080/03605300601188748.

[10]

Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal., 31 (1999), 1-18. doi: 10.1137/S0036141099352844.

[11]

Y. Du, "Order Structure and Topological Methods in Nonlinear Partial Differential Equations," Vol. 1. Maximum Principles and Applications, Series in Partial Differential Equations and Applications, 2. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.

[12]

S. Dumont, L. Dupaigne, O. Goubet and V. D. Rădulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Advanced Nonlinear Studies, 7 (2007), 271-298.

[13]

J. García - Melián, R. Letelier-Albornoz and J. Sabina de Lis, Uniqueness and asymptotic behavior for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. Soc., 129 (2001), 3593-3602. doi: 10.1090/S0002-9939-01-06229-3.

[14]

J. García - Melián, Boundary behavior of large solutions to elliptic equations with singular weights, Nonlinear Anal., 67 (2007), 818-826. doi: 10.1016/j.na.2006.06.041.

[15]

J. García - Melián, Uniqueness of positive solutions for a boundary blow-up problem, J. Math. Anal. Appl., 360 (2009), 530-536. doi: 10.1016/j.jmaa.2009.06.077.

[16]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 3nd edition, Springer - Verlag, Berlin, 1998.

[17]

F. Gladiali and G. Porru, Estimates for explosive solutions to $p$-Laplace equations, Progress in partial diffrential equations, Vol. 1 (Pont-à-Mousson, 1997), 117-127, Pitman Res. Notes Math. Ser., 383, Longman, Harlow, 1998.

[18]

S. Huang, Q. Tian, S. Zhang and J. Xi, A second order estimate for blow-up solutions of elliptic equations, Nonlinear Anal., 74 (2011), 2342-2350. doi: 10.1016/j.na.2010.11.037.

[19]

J. B. Keller, On solutions of $\Delta u=f(u)$, Commun. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.

[20]

A. V. Lair, A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations, J. Math. Anal. Appl., 240 (1999), 205-218. doi: 10.1006/jmaa.1999.6609.

[21]

A. C. Lazer and P. J. McKenna, Asymptotic behavior of solutions of boundary blowup problems, Differential Integral Equations, 7 (1994), 1001-1019.

[22]

C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, 245-272.

[23]

J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Diff. Equations, 224 (2006), 385-439. doi: 10.1016/j.jde.2005.08.008.

[24]

J. López-Gómez, Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst. 2007, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., 677-686.

[25]

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.

[26]

M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equations, J. Evol. Equations, 3 (2003), 637-652. doi: 10.1007/s00028-003-0122-y.

[27]

V. Maric, "Regular Variation and Differential Equations, '' Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952.

[28]

A. Mohammed, Boundary asymtotic and uniqueness of solutions to the p-Laplacian with infinite boundary value, J. Math. Anal. Appl., 325 (2007), 480-489. doi: 10.1016/j.jmaa.2006.02.008.

[29]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.

[30]

S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes," Springer-Verlag, New York, Berlin, 1987.

[31]

R. Seneta, "Regular Varying Functions," Lecture Notes in Math., vol. 508, Springer-Verlag, 1976. doi: 10.1007/BFb0079658.

[32]

S. Tao and Z. Zhang, On the existence of explosive solutions for semilinear elliptic problems, Nonlinear Anal., 48 (2002), 1043-1050. doi: 10.1016/S0362-546X(00)00233-9.

[33]

Z. Xie, Uniqueness and blow-up rate of large solutions for elliptic equation $-\Delta u =\lambda u-b(x)h(u)$, J. Diff. Equations, 247 (2009), 344-363. doi: 10.1016/j.jde.2009.04.001.

[34]

Z. Zhang, A remark on the existence of explosive solutions for a class of semilinear elliptic equations, Nonlinear Anal., 41 (2000), 143-148. doi: 10.1016/S0362-546X(98)00270-3.

[35]

Z. Zhang, Boundary behavior of solutions to some singular elliptic boundary value problems, Nonlinear Anal., 69 (2008), 2293-2302. doi: 10.1016/j.na.2007.03.034.

[36]

Z. Zhang, X. Li and Y. Zhao, Boundary behavior of solutions to singular boundary value problems for nonlinear elliptic equations, Advanced Nonlinear Studies, 10 (2010), 249-261.

[37]

Z. Zhang, Y. Ma, L. Mi and X. Li, Blow-up rates of large solutions for elliptic equations, J. Diff. Equations, 249 (2010), 180-199. doi: 10.1016/j.jde.2010.02.019.

show all references

References:
[1]

C. Anedda and G. Porru, Second order estimates for boundary blow-up solutions of elliptic equations, Discrete Contin. Dyn. Syst., (Suppl.) (2007), 54-63.

[2]

C. Anedda and G. Porru, Boundary behaviour for solutions of boundary blow-up problems in a borderline case, J. Math. Anal. Appl., 352 (2009), 35-47. doi: 10.1016/j.jmaa.2008.02.042.

[3]

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.

[4]

C. Bandle, Asymptotic behavior of large solutions of quasilinear elliptic problems, Z. angew. Math. Phys., 54 (2003), 731-738. doi: 10.1007/s00033-003-3207-0.

[5]

N. H. Bingham, C. M. Goldie and J. L. Teugels, "Regular Variation," Encyclopedia of Mathematics and its Applications 27, Cambridge University Press, Cambridge, 1987.

[6]

F. Cîrstea and V. D. Rădulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Acad. Sci. Paris, Sér. I, 335 (2002), 447-452. doi: 10.1112/S1631-073X(02)02523-7/FLA.

[7]

F. Cirstea and Y. Du, General uniqueness results and variation speed for blow-up solutions of elliptic equations, Proc. London Math. Soc., 91 (2005), 459-482. doi: 10.1112/S0024611505015273.

[8]

F. Cîrstea, Elliptic equations with competing rapidly varying nonlinearities and boundary blow-up, Advances in Differential Equations, 12 (2007), 995-1030.

[9]

H. Dong, S. Kim and M. Safonov, On uniqueness of boundary blow-up solutions of a class of nonlinear elliptic equations, Comm. Partial Diff. Equations, 33 (2008), 177-188. doi: 10.1080/03605300601188748.

[10]

Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal., 31 (1999), 1-18. doi: 10.1137/S0036141099352844.

[11]

Y. Du, "Order Structure and Topological Methods in Nonlinear Partial Differential Equations," Vol. 1. Maximum Principles and Applications, Series in Partial Differential Equations and Applications, 2. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.

[12]

S. Dumont, L. Dupaigne, O. Goubet and V. D. Rădulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Advanced Nonlinear Studies, 7 (2007), 271-298.

[13]

J. García - Melián, R. Letelier-Albornoz and J. Sabina de Lis, Uniqueness and asymptotic behavior for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. Soc., 129 (2001), 3593-3602. doi: 10.1090/S0002-9939-01-06229-3.

[14]

J. García - Melián, Boundary behavior of large solutions to elliptic equations with singular weights, Nonlinear Anal., 67 (2007), 818-826. doi: 10.1016/j.na.2006.06.041.

[15]

J. García - Melián, Uniqueness of positive solutions for a boundary blow-up problem, J. Math. Anal. Appl., 360 (2009), 530-536. doi: 10.1016/j.jmaa.2009.06.077.

[16]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 3nd edition, Springer - Verlag, Berlin, 1998.

[17]

F. Gladiali and G. Porru, Estimates for explosive solutions to $p$-Laplace equations, Progress in partial diffrential equations, Vol. 1 (Pont-à-Mousson, 1997), 117-127, Pitman Res. Notes Math. Ser., 383, Longman, Harlow, 1998.

[18]

S. Huang, Q. Tian, S. Zhang and J. Xi, A second order estimate for blow-up solutions of elliptic equations, Nonlinear Anal., 74 (2011), 2342-2350. doi: 10.1016/j.na.2010.11.037.

[19]

J. B. Keller, On solutions of $\Delta u=f(u)$, Commun. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.

[20]

A. V. Lair, A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations, J. Math. Anal. Appl., 240 (1999), 205-218. doi: 10.1006/jmaa.1999.6609.

[21]

A. C. Lazer and P. J. McKenna, Asymptotic behavior of solutions of boundary blowup problems, Differential Integral Equations, 7 (1994), 1001-1019.

[22]

C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, 245-272.

[23]

J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Diff. Equations, 224 (2006), 385-439. doi: 10.1016/j.jde.2005.08.008.

[24]

J. López-Gómez, Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst. 2007, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., 677-686.

[25]

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.

[26]

M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equations, J. Evol. Equations, 3 (2003), 637-652. doi: 10.1007/s00028-003-0122-y.

[27]

V. Maric, "Regular Variation and Differential Equations, '' Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952.

[28]

A. Mohammed, Boundary asymtotic and uniqueness of solutions to the p-Laplacian with infinite boundary value, J. Math. Anal. Appl., 325 (2007), 480-489. doi: 10.1016/j.jmaa.2006.02.008.

[29]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.

[30]

S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes," Springer-Verlag, New York, Berlin, 1987.

[31]

R. Seneta, "Regular Varying Functions," Lecture Notes in Math., vol. 508, Springer-Verlag, 1976. doi: 10.1007/BFb0079658.

[32]

S. Tao and Z. Zhang, On the existence of explosive solutions for semilinear elliptic problems, Nonlinear Anal., 48 (2002), 1043-1050. doi: 10.1016/S0362-546X(00)00233-9.

[33]

Z. Xie, Uniqueness and blow-up rate of large solutions for elliptic equation $-\Delta u =\lambda u-b(x)h(u)$, J. Diff. Equations, 247 (2009), 344-363. doi: 10.1016/j.jde.2009.04.001.

[34]

Z. Zhang, A remark on the existence of explosive solutions for a class of semilinear elliptic equations, Nonlinear Anal., 41 (2000), 143-148. doi: 10.1016/S0362-546X(98)00270-3.

[35]

Z. Zhang, Boundary behavior of solutions to some singular elliptic boundary value problems, Nonlinear Anal., 69 (2008), 2293-2302. doi: 10.1016/j.na.2007.03.034.

[36]

Z. Zhang, X. Li and Y. Zhao, Boundary behavior of solutions to singular boundary value problems for nonlinear elliptic equations, Advanced Nonlinear Studies, 10 (2010), 249-261.

[37]

Z. Zhang, Y. Ma, L. Mi and X. Li, Blow-up rates of large solutions for elliptic equations, J. Diff. Equations, 249 (2010), 180-199. doi: 10.1016/j.jde.2010.02.019.

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