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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 & 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., (2007), 54.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2008.02.042.  Google Scholar

[3]

C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior,, J. Anal. Math., 58 (1992), 9.  doi: 10.1007/BF02790355.  Google Scholar

[4]

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

[5]

N. H. Bingham, C. M. Goldie and J. L. Teugels, "Regular Variation,", Encyclopedia of Mathematics and its Applications 27, (1987).   Google Scholar

[6]

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

[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.  doi: 10.1112/S0024611505015273.  Google Scholar

[8]

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

[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.  doi: 10.1080/03605300601188748.  Google Scholar

[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.  doi: 10.1137/S0036141099352844.  Google Scholar

[11]

Y. Du, "Order Structure and Topological Methods in Nonlinear Partial Differential Equations,", Vol. 1. Maximum Principles and Applications, (2006).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1090/S0002-9939-01-06229-3.  Google Scholar

[14]

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

[15]

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

[16]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 3nd edition, (1998).   Google Scholar

[17]

F. Gladiali and G. Porru, Estimates for explosive solutions to $p$-Laplace equations,, Progress in partial diffrential equations, (1997), 117.   Google Scholar

[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.  doi: 10.1016/j.na.2010.11.037.  Google Scholar

[19]

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

[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.  doi: 10.1006/jmaa.1999.6609.  Google Scholar

[21]

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

[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), (1974), 245.   Google Scholar

[23]

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

[24]

J. López-Gómez, Uniqueness of radially symmetric large solutions,, Discrete Contin. Dyn. Syst. 2007, (2007), 677.   Google Scholar

[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. Poincar\'e Anal. Non Lin\'eaire, 14 (1997), 237.   Google Scholar

[26]

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

[27]

V. Maric, "Regular Variation and Differential Equations, '', Lecture Notes in Math., (1726).  doi: 10.1007/BFb0103952.  Google Scholar

[28]

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

[29]

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

[30]

S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes,", Springer-Verlag, (1987).   Google Scholar

[31]

R. Seneta, "Regular Varying Functions,", Lecture Notes in Math., (1976).  doi: 10.1007/BFb0079658.  Google Scholar

[32]

S. Tao and Z. Zhang, On the existence of explosive solutions for semilinear elliptic problems,, \emph{On the existence of explosive solutions for semilinear elliptic problems}, ().  doi: 10.1016/S0362-546X(00)00233-9.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.04.001.  Google Scholar

[34]

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

[35]

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

[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.   Google Scholar

[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.  doi: 10.1016/j.jde.2010.02.019.  Google Scholar

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., (2007), 54.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2008.02.042.  Google Scholar

[3]

C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior,, J. Anal. Math., 58 (1992), 9.  doi: 10.1007/BF02790355.  Google Scholar

[4]

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

[5]

N. H. Bingham, C. M. Goldie and J. L. Teugels, "Regular Variation,", Encyclopedia of Mathematics and its Applications 27, (1987).   Google Scholar

[6]

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

[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.  doi: 10.1112/S0024611505015273.  Google Scholar

[8]

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

[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.  doi: 10.1080/03605300601188748.  Google Scholar

[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.  doi: 10.1137/S0036141099352844.  Google Scholar

[11]

Y. Du, "Order Structure and Topological Methods in Nonlinear Partial Differential Equations,", Vol. 1. Maximum Principles and Applications, (2006).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1090/S0002-9939-01-06229-3.  Google Scholar

[14]

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

[15]

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

[16]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 3nd edition, (1998).   Google Scholar

[17]

F. Gladiali and G. Porru, Estimates for explosive solutions to $p$-Laplace equations,, Progress in partial diffrential equations, (1997), 117.   Google Scholar

[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.  doi: 10.1016/j.na.2010.11.037.  Google Scholar

[19]

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

[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.  doi: 10.1006/jmaa.1999.6609.  Google Scholar

[21]

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

[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), (1974), 245.   Google Scholar

[23]

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

[24]

J. López-Gómez, Uniqueness of radially symmetric large solutions,, Discrete Contin. Dyn. Syst. 2007, (2007), 677.   Google Scholar

[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. Poincar\'e Anal. Non Lin\'eaire, 14 (1997), 237.   Google Scholar

[26]

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

[27]

V. Maric, "Regular Variation and Differential Equations, '', Lecture Notes in Math., (1726).  doi: 10.1007/BFb0103952.  Google Scholar

[28]

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

[29]

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

[30]

S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes,", Springer-Verlag, (1987).   Google Scholar

[31]

R. Seneta, "Regular Varying Functions,", Lecture Notes in Math., (1976).  doi: 10.1007/BFb0079658.  Google Scholar

[32]

S. Tao and Z. Zhang, On the existence of explosive solutions for semilinear elliptic problems,, \emph{On the existence of explosive solutions for semilinear elliptic problems}, ().  doi: 10.1016/S0362-546X(00)00233-9.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.04.001.  Google Scholar

[34]

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

[35]

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

[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.   Google Scholar

[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.  doi: 10.1016/j.jde.2010.02.019.  Google Scholar

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