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

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 & 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. doi: 10.1016/j.jde.2011.09.033. Google Scholar
[2]	H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems,, Math. Z., 150 (1976), 281. doi: 10.1007/BF01221152. Google Scholar
[3]	C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior,, J. Analyse Math., 58 (1992), 9. doi: 10.1007/BF02790355. Google Scholar
[4]	C. Bandle and E. Giarrusso, Boundary blow-up for semilinear elliptic equations with nonlinear gradient term,, Adv. Differential Equations, 1 (1996), 133. Google Scholar
[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. doi: 10.1016/S0022-247X(03)00058-1. Google Scholar
[6]	Y. Chen and M. Wang, Large solutions for quasilinear elliptic equation with nonlinear gradient term,, Nonlinear Anal.: Real World Appl., 12 (2011), 455. doi: 10.1016/j.nonrwa.2010.06.031. Google Scholar
[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. Google Scholar
[8]	F. Cîrstea and V. Rădulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion,, C. R. Acad. Sci. Paris, 335 (2002), 447. doi: 10.1016/S1631-073X(02)02503-7/FLA. Google Scholar
[9]	F. Cîrstea and V. Rădulescu, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach,, Asymptotic Anal., 46 (2006), 275. Google Scholar
[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. doi: 10.1112/S0024611505015273. Google Scholar
[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. doi: 10.4418/2010.65.2.8. Google Scholar
[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. doi: 10.4171/ZAA/1406. Google Scholar
[13]	E. Giarrusso and G. Porru, Problems for elliptic singular equations with a gradient term,, Nonlinear Anal., 65 (2006), 107. doi: 10.1016/j.na.2005.08.007. Google Scholar
[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. doi: 10.1016/S0764-4442(00)01707-9/FLA. Google Scholar
[15]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 3nd edition,, Springer - Verlag, (1998). Google Scholar
[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. doi: 10.1016/j.amc.2006.03.029. Google Scholar
[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. doi: 10.1016/j.jde.2011.08.031. Google Scholar
[18]	J. B. Keller, On solutions of $\Delta u=f(u)$,, Commun. Pure Appl. Math., 10 (1957), 503. doi: 10.1002/cpa.3160100402. Google Scholar
[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. doi: 10.1006/S 0161-1712<99>22869-4. 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]	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. doi: 10.1007/BF01442856. Google Scholar
[22]	T. Leonori, Large solutions for a class of nonlinear elliptic equations with gradient terms,, Adv. Nonlinear Studies, 7 (2007), 237. Google Scholar
[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. doi: 10.1137/070681363. Google Scholar
[24]	V. Maric, "Regular Variation and Differential Equations,'', Lecture Notes in Math., (1726). doi: 10.1007/BFb0103952. Google Scholar
[25]	R. Osserman, On the inequality $\Delta u\geq f(u)$,, Pacific J. Math., 7 (1957), 1641. Google Scholar
[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. Google Scholar
[27]	S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes,'', Springer-Verlag, (1987). Google Scholar
[28]	Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms,, J. Differential Equations, 228 (2006), 661. doi: 10.1016/j.jde.2006.02.003. Google Scholar
[29]	Z. Zhang, Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms,, Nonlinear Anal., 73 (2010), 3348. doi: 10.1016/j.na.2010.07.017. Google Scholar
[30]	Z. Zhang, Nonlinear elliptic equations with singular boundary conditions,, J. Math. Anal. Appl., 216 (1997), 390. doi: 10.1006//jmaa.1997.5635. Google Scholar
[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. doi: 10.1016/j.jde.2010.02.019. Google Scholar

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. doi: 10.1016/j.jde.2011.09.033. Google Scholar
[2]	H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems,, Math. Z., 150 (1976), 281. doi: 10.1007/BF01221152. Google Scholar
[3]	C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior,, J. Analyse Math., 58 (1992), 9. doi: 10.1007/BF02790355. Google Scholar
[4]	C. Bandle and E. Giarrusso, Boundary blow-up for semilinear elliptic equations with nonlinear gradient term,, Adv. Differential Equations, 1 (1996), 133. Google Scholar
[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. doi: 10.1016/S0022-247X(03)00058-1. Google Scholar
[6]	Y. Chen and M. Wang, Large solutions for quasilinear elliptic equation with nonlinear gradient term,, Nonlinear Anal.: Real World Appl., 12 (2011), 455. doi: 10.1016/j.nonrwa.2010.06.031. Google Scholar
[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. Google Scholar
[8]	F. Cîrstea and V. Rădulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion,, C. R. Acad. Sci. Paris, 335 (2002), 447. doi: 10.1016/S1631-073X(02)02503-7/FLA. Google Scholar
[9]	F. Cîrstea and V. Rădulescu, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach,, Asymptotic Anal., 46 (2006), 275. Google Scholar
[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. doi: 10.1112/S0024611505015273. Google Scholar
[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. doi: 10.4418/2010.65.2.8. Google Scholar
[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. doi: 10.4171/ZAA/1406. Google Scholar
[13]	E. Giarrusso and G. Porru, Problems for elliptic singular equations with a gradient term,, Nonlinear Anal., 65 (2006), 107. doi: 10.1016/j.na.2005.08.007. Google Scholar
[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. doi: 10.1016/S0764-4442(00)01707-9/FLA. Google Scholar
[15]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 3nd edition,, Springer - Verlag, (1998). Google Scholar
[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. doi: 10.1016/j.amc.2006.03.029. Google Scholar
[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. doi: 10.1016/j.jde.2011.08.031. Google Scholar
[18]	J. B. Keller, On solutions of $\Delta u=f(u)$,, Commun. Pure Appl. Math., 10 (1957), 503. doi: 10.1002/cpa.3160100402. Google Scholar
[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. doi: 10.1006/S 0161-1712<99>22869-4. 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]	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. doi: 10.1007/BF01442856. Google Scholar
[22]	T. Leonori, Large solutions for a class of nonlinear elliptic equations with gradient terms,, Adv. Nonlinear Studies, 7 (2007), 237. Google Scholar
[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. doi: 10.1137/070681363. Google Scholar
[24]	V. Maric, "Regular Variation and Differential Equations,'', Lecture Notes in Math., (1726). doi: 10.1007/BFb0103952. Google Scholar
[25]	R. Osserman, On the inequality $\Delta u\geq f(u)$,, Pacific J. Math., 7 (1957), 1641. Google Scholar
[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. Google Scholar
[27]	S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes,'', Springer-Verlag, (1987). Google Scholar
[28]	Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms,, J. Differential Equations, 228 (2006), 661. doi: 10.1016/j.jde.2006.02.003. Google Scholar
[29]	Z. Zhang, Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms,, Nonlinear Anal., 73 (2010), 3348. doi: 10.1016/j.na.2010.07.017. Google Scholar
[30]	Z. Zhang, Nonlinear elliptic equations with singular boundary conditions,, J. Math. Anal. Appl., 216 (1997), 390. doi: 10.1006//jmaa.1997.5635. Google Scholar
[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. doi: 10.1016/j.jde.2010.02.019. Google Scholar

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