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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

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), 886-914. 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), 281-295. 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), 9-24. 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), 133-150.  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), 123-132. 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), 455-463. 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), 717-737.  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, Sér. I, 335 (2002), 447-452. 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), 275-298.  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), 459-482. 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), 55-68. 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), 219-234. 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), 107-128. 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), 777-782. 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, Berlin, 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), 13-23. 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), 3297-3328. 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), 503-510. 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), 869-883. 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), 205-218. 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), 583-630. 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), 237-269.  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), 1295-1327. doi: 10.1137/070681363.  Google Scholar

[24]

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

[25]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.  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), 351-378.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[30]

Z. Zhang, Nonlinear elliptic equations with singular boundary conditions, J. Math. Anal. Appl., 216 (1997), 390-397. 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), 180-199. 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), 886-914. 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), 281-295. 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), 9-24. 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), 133-150.  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), 123-132. 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), 455-463. 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), 717-737.  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, Sér. I, 335 (2002), 447-452. 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), 275-298.  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), 459-482. 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), 55-68. 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), 219-234. 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), 107-128. 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), 777-782. 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, Berlin, 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), 13-23. 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), 3297-3328. 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), 503-510. 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), 869-883. 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), 205-218. 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), 583-630. 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), 237-269.  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), 1295-1327. doi: 10.1137/070681363.  Google Scholar

[24]

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

[25]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.  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), 351-378.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[30]

Z. Zhang, Nonlinear elliptic equations with singular boundary conditions, J. Math. Anal. Appl., 216 (1997), 390-397. 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), 180-199. doi: 10.1016/j.jde.2010.02.019.  Google Scholar

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