Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior

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May 2013, 12(3): 1381-1392. doi: 10.3934/cpaa.2013.12.1381

Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior

Zhijun Zhang ^1,

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

Received April 2012 Revised July 2012 Published September 2012

Abstract
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In this paper, we study the existence and boundary behavior of solutions to boundary blow-up elliptic problems \begin{eqnarray*} \triangle u =b(x)f(u)(1+|\nabla u|^q), u\geq 0, \ x\in \Omega,\ u|_{\partial \Omega}=\infty, \end{eqnarray*} where $\Omega$ is a bounded domain with smooth boundary in $\mathbb R^N$, $q\in (0, 2]$, $b \in C^{\alpha}(\bar{\Omega})$ which is positive in $\Omega$, may be vanishing on the boundary, and $f$ is normalised regularly varying at infinity with positive index $p$ and $p+q>1$.

Keywords: Semilinear elliptic equations, boundary behavior of solutions, boundary blow-up, a gradient term, uniqueness., existence.

Mathematics Subject Classification: Primary: 35J25, 35J65, 35J6.

Citation: Zhijun Zhang. Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1381-1392. doi: 10.3934/cpaa.2013.12.1381

References:

[1]	S. Alarcón, J. García-Melián and A. Quaas, Keller-Osserman type conditions for some elliptic problems with gradient terms, J. Differential Equations, 252 (2012), 886-914. doi: 10.1016/j.jde.2011.09.033.
[2]	H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295. doi: 10.1007/BF01221152.
[3]	C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. Analyse Math., 58 (1992), 9-24. doi: 10.1007/BF02790355.
[4]	C. Bandle and E. Giarrusso, Boundary blow-up for semilinear elliptic equations with nonlinear gradient term, Adv. Differential Equations, 1 (1996), 133-150.
[5]	E. B. Castillo and R. L. Albornoz, Local gradient estimates and existence of blow-up solutions to a class of quasilinear elliptic equations, J. Math. Anal. Appl., 280 (2003), 123-132. doi: 10.1016/S0022-247X(03)00058-1.
[6]	Y. Chen and M. Wang, Large solutions for quasilinear elliptic equation with nonlinear gradient term, Nonlinear Anal.: Real World Appl., 12 (2011), 455-463. doi: 10.1016/j.nonrwa.2010.06.031.
[7]	Y. Chen and M. Wang, Boundary blow-up solutions for elliptic equations with gradient terms and singular weights : existence, asymptotic behaviour and uniqueness, Proc. Roy. Soc. Edinb., 141A (2011), 717-737.
[8]	F. Cîrstea and V. 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: 10.1016/S1631-073X(02)02503-7/FLA.
[9]	F. Cîrstea and V. Rădulescu, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach, Asymptotic Anal., 46 (2006), 275-298.
[10]	F. Cîrstea 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.
[11]	V. Ferone, Boundary blow-up for nonlinear elliptic equations with general growth in the gradient: an approach via symmetrisation, Le Matematiche, 65 (2010), 55-68. doi: 10.4418/2010.65.2.8.
[12]	V. Ferone, E. Giarrusso, B. Messano and M. R. Posteraro, Estimates for blow-up solutions to nonlinear elliptic equations with $p$-growth in the gradient, Z. Anal. Anwend., 29 (2010), 219-234. doi: 10.4171/ZAA/1406.
[13]	E. Giarrusso and G. Porru, Problems for elliptic singular equations with a gradient term, Nonlinear Anal., 65 (2006), 107-128. doi: 10.1016/j.na.2005.08.007.
[14]	E. Giarrusso, Asymptotic behavior of large solutions of an elliptic quasilinear equation in a borderline case, C.R. Acad. Sci. Paris Ser. I, 331 (2000), 777-782. doi: 10.1016/S0764-4442(00)01707-9/FLA.
[15]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 3nd edition, Springer - Verlag, Berlin, 1998.
[16]	V. Goncalves, A. Roncalli, Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput., 182 (2006), 13-23. doi: 10.1016/j.amc.2006.03.029.
[17]	S. Huang, W. Li, Q. Tian and C. Mu, Large solution to nonlinear elliptic equation with nonlinear gradient terms, J. Diff. Equations, 251 (2011), 3297-3328. doi: 10.1016/j.jde.2011.08.031.
[18]	J. B. Keller, On solutions of $\Delta u=f(u)$, Commun. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.
[19]	A. V. Lair and A. W. Wood, Large solutions of semilinear elliptic equations with nonlinear gradient terms, Int. J. Math. Math. Sci., 22 (1999), 869-883. doi: 10.1006/S 0161-1712<99>22869-4.
[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]	J. M. Lasry and P. L. Lions, Nonlinear elliptic equations with singular boundary Conditions and stochastic control with state constrains, Math. Ann., 283 (1989), 583-630. doi: 10.1007/BF01442856.
[22]	T. Leonori, Large solutions for a class of nonlinear elliptic equations with gradient terms, Adv. Nonlinear Studies, 7 (2007), 237-269.
[23]	T. Leonori and A. Porretta, The boundary behavior of blow-up solutions related to a stochastic control problem with state constraint, SIAM J. Math. Anal., 39 (2007), 1295-1327. doi: 10.1137/070681363.
[24]	V. Maric, "Regular Variation and Differential Equations,'' Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952.
[25]	R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.
[26]	A. Porretta and L. Véron, Asymptotic behaviour for the gradient of large solutions to some nonlinear elliptic equations, Adv. Nonlinear Studies, 6 (2006), 351-378.
[27]	S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes,'' Springer-Verlag, New York, Berlin, 1987.
[28]	Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms, J. Differential Equations, 228 (2006), 661-684. doi: 10.1016/j.jde.2006.02.003.
[29]	Z. Zhang, Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms, Nonlinear Anal., 73 (2010), 3348-3363. doi: 10.1016/j.na.2010.07.017.
[30]	Z. Zhang, Nonlinear elliptic equations with singular boundary conditions, J. Math. Anal. Appl., 216 (1997), 390-397. doi: 10.1006//jmaa.1997.5635.
[31]	Z. Zhang, Y. Ma, L. Mi and X. Li, Blow-up rates of large solutions for elliptic equations, J. Differential Equations, 249 (2010), 180-199. doi: 10.1016/j.jde.2010.02.019.

show all references

References:

[1]	S. Alarcón, J. García-Melián and A. Quaas, Keller-Osserman type conditions for some elliptic problems with gradient terms, J. Differential Equations, 252 (2012), 886-914. doi: 10.1016/j.jde.2011.09.033.
[2]	H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295. doi: 10.1007/BF01221152.
[3]	C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. Analyse Math., 58 (1992), 9-24. doi: 10.1007/BF02790355.
[4]	C. Bandle and E. Giarrusso, Boundary blow-up for semilinear elliptic equations with nonlinear gradient term, Adv. Differential Equations, 1 (1996), 133-150.
[5]	E. B. Castillo and R. L. Albornoz, Local gradient estimates and existence of blow-up solutions to a class of quasilinear elliptic equations, J. Math. Anal. Appl., 280 (2003), 123-132. doi: 10.1016/S0022-247X(03)00058-1.
[6]	Y. Chen and M. Wang, Large solutions for quasilinear elliptic equation with nonlinear gradient term, Nonlinear Anal.: Real World Appl., 12 (2011), 455-463. doi: 10.1016/j.nonrwa.2010.06.031.
[7]	Y. Chen and M. Wang, Boundary blow-up solutions for elliptic equations with gradient terms and singular weights : existence, asymptotic behaviour and uniqueness, Proc. Roy. Soc. Edinb., 141A (2011), 717-737.
[8]	F. Cîrstea and V. 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: 10.1016/S1631-073X(02)02503-7/FLA.
[9]	F. Cîrstea and V. Rădulescu, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach, Asymptotic Anal., 46 (2006), 275-298.
[10]	F. Cîrstea 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.
[11]	V. Ferone, Boundary blow-up for nonlinear elliptic equations with general growth in the gradient: an approach via symmetrisation, Le Matematiche, 65 (2010), 55-68. doi: 10.4418/2010.65.2.8.
[12]	V. Ferone, E. Giarrusso, B. Messano and M. R. Posteraro, Estimates for blow-up solutions to nonlinear elliptic equations with $p$-growth in the gradient, Z. Anal. Anwend., 29 (2010), 219-234. doi: 10.4171/ZAA/1406.
[13]	E. Giarrusso and G. Porru, Problems for elliptic singular equations with a gradient term, Nonlinear Anal., 65 (2006), 107-128. doi: 10.1016/j.na.2005.08.007.
[14]	E. Giarrusso, Asymptotic behavior of large solutions of an elliptic quasilinear equation in a borderline case, C.R. Acad. Sci. Paris Ser. I, 331 (2000), 777-782. doi: 10.1016/S0764-4442(00)01707-9/FLA.
[15]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 3nd edition, Springer - Verlag, Berlin, 1998.
[16]	V. Goncalves, A. Roncalli, Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput., 182 (2006), 13-23. doi: 10.1016/j.amc.2006.03.029.
[17]	S. Huang, W. Li, Q. Tian and C. Mu, Large solution to nonlinear elliptic equation with nonlinear gradient terms, J. Diff. Equations, 251 (2011), 3297-3328. doi: 10.1016/j.jde.2011.08.031.
[18]	J. B. Keller, On solutions of $\Delta u=f(u)$, Commun. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.
[19]	A. V. Lair and A. W. Wood, Large solutions of semilinear elliptic equations with nonlinear gradient terms, Int. J. Math. Math. Sci., 22 (1999), 869-883. doi: 10.1006/S 0161-1712<99>22869-4.
[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]	J. M. Lasry and P. L. Lions, Nonlinear elliptic equations with singular boundary Conditions and stochastic control with state constrains, Math. Ann., 283 (1989), 583-630. doi: 10.1007/BF01442856.
[22]	T. Leonori, Large solutions for a class of nonlinear elliptic equations with gradient terms, Adv. Nonlinear Studies, 7 (2007), 237-269.
[23]	T. Leonori and A. Porretta, The boundary behavior of blow-up solutions related to a stochastic control problem with state constraint, SIAM J. Math. Anal., 39 (2007), 1295-1327. doi: 10.1137/070681363.
[24]	V. Maric, "Regular Variation and Differential Equations,'' Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952.
[25]	R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.
[26]	A. Porretta and L. Véron, Asymptotic behaviour for the gradient of large solutions to some nonlinear elliptic equations, Adv. Nonlinear Studies, 6 (2006), 351-378.
[27]	S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes,'' Springer-Verlag, New York, Berlin, 1987.
[28]	Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms, J. Differential Equations, 228 (2006), 661-684. doi: 10.1016/j.jde.2006.02.003.
[29]	Z. Zhang, Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms, Nonlinear Anal., 73 (2010), 3348-3363. doi: 10.1016/j.na.2010.07.017.
[30]	Z. Zhang, Nonlinear elliptic equations with singular boundary conditions, J. Math. Anal. Appl., 216 (1997), 390-397. doi: 10.1006//jmaa.1997.5635.
[31]	Z. Zhang, Y. Ma, L. Mi and X. Li, Blow-up rates of large solutions for elliptic equations, J. Differential Equations, 249 (2010), 180-199. doi: 10.1016/j.jde.2010.02.019.

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