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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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