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Exact boundary behavior of large solutions to semilinear elliptic equations with a nonlinear gradient term. (English) Zbl 1437.35364

Summary: This paper is concerned with exact boundary behavior of large solutions to semilinear elliptic equations \(\Delta u = b(x)f(u)(C_0 + |\nabla u|^q)\), \(x \in \Omega \), where \(\Omega\) is a bounded domain with a smooth boundary in \(\mathbb{R}^N\), \(C_0 \geqslant 0\), \(q \in [0, 2)\), \(b \in C_{\mathrm{loc}}^\alpha(\Omega)\) is positive in \(\Omega \), but may be vanishing or appropriately singular on the boundary, \(f \in C[0, \infty )\), \(f(0) = 0\), and \(f\) is strictly increasing on \([0, \infty )\) (or \(f \in C( \mathbb{R} )\), \(f(s) > 0\;\, \forall s \in \mathbb{R}\), \(f\) is strictly increasing on \(\mathbb{R} \)). We show unified boundary behavior of such solutions to the problem under a new structure condition on \(f\).

MSC:

35J62 Quasilinear elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
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[1] Bandle, C.; Giarrusso, E., Boundary blow-up for semilinear elliptic equations with nonlinear gradient term, Adv Differential Equations, 1, 133-150 (1996) · Zbl 0840.35034
[2] Bandle, C.; Marcus, M., Large solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behavior, J Anal Math, 58, 9-24 (1992) · Zbl 0802.35038
[3] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge: Cambridge University Press, Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge · Zbl 0617.26001
[4] Castillo, E. B.; Albornoz, R. L., Local gradient estimates and existence of blow-up solutions to a class of quasilinear elliptic equations, J Math Anal Appl, 280, 123-132 (2003) · Zbl 1284.35190
[5] Chen, Y. J.; Pang, P. Y H.; Wang, M. X., Blow-up rates and uniqueness of large solutions for elliptic equations with nonlinear gradient term and singular or degenerate weights, Manuscripta Math, 141, 171-193 (2013) · Zbl 1268.35033
[6] Chen, Y. J.; Wang, M. X., Large solutions for quasilinear elliptic equation with nonlinear gradient term, Nonlinear Anal Real World Appl, 12, 455-463 (2011) · Zbl 1205.35106
[7] Chen, Y. J.; Wang, M. X., Boundary blow-up solutions for elliptic equations with gradient terms and singular weights: Existence, asymptotic behaviour and uniqueness, Proc Roy Soc Edinburgh Sect A, 141, 717-737 (2011) · Zbl 1263.35078
[8] Cîrstea, F.; Du, Y. H., General uniqueness results and variation speed for blow-up solutions of elliptic equations, Proc Lond Math Soc (3), 91, 459-482 (2005) · Zbl 1108.35068
[9] Cîrstea, F.; Rădulescu, V., Uniqueness of the blow-up boundary solution of logistic equations with absorption, C R Acad Sci Paris Ser I, 335, 447-452 (2002) · Zbl 1183.35124
[10] Del Pino, M.; Letelier, R., The influence of domain geometry in boundary blow-up elliptic problems, Nonlinear Anal, 48, 897-904 (2002) · Zbl 1142.35431
[11] Diaz, G.; Letelier, R., Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal, 20, 97-125 (1993) · Zbl 0793.35028
[12] Dong, H. J.; Kim, S.; Safonov, M., On uniqueness of boundary blow-up solutions of a class of nonlinear elliptic equations, Comm Partial Differential Equations, 33, 177-188 (2008) · Zbl 1153.35039
[13] Du, Y. H., Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Volume 1 (2006), Maximum Principles and Applications. Series in Partial Differential Equations and Applications, vol. 2. Hackensack: World Scientific, Maximum Principles and Applications. Series in Partial Differential Equations and Applications, vol. 2. Hackensack · Zbl 1202.35043
[14] Du, Y. H.; Guo, Z. M., Boundary blow-up solutions and their applications in quasilinear elliptic equations, J Anal Math, 89, 277-302 (2003) · Zbl 1162.35028
[15] Ferone, V., Boundary blow-up for nonlinear elliptic equations with general growth in the gradient: An approach via symmetrisation, Matematiche (Catania), 65, 55-68 (2010) · Zbl 1219.35096
[16] García-Melián, J., Boundary behavior of large solutions to elliptic equations with singular weights, Nonlinear Anal, 67, 818-826 (2007) · Zbl 1143.35054
[17] Giarrusso, E., Asymptotic behavior of large solutions of an elliptic quasilinear equation in a borderline case, C R Acad Sci Paris Ser I, 331, 777-782 (2000) · Zbl 0966.35041
[18] Giarrusso, E.; Marras, M.; Porru, G., Second order estimates for boundary blowup solutions of quasilinear elliptic equations, J Math Anal Appl, 424, 444-459 (2015) · Zbl 1308.35103
[19] Giarrusso, E.; Porru, G., Second order estimates for boundary blow-up solutions of elliptic equations with an additive gradient term, Nonlinear Anal, 129, 160-172 (2015) · Zbl 1328.35059
[20] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 3rd ed (1998), Berlin: Springer-Verlag, Berlin · Zbl 0691.35001
[21] Goncalves, V.; Roncalli, A., Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl Math Comput, 182, 13-23 (2006) · Zbl 1186.35079
[22] Guo, Z. M.; Webb, J. R L., Structure of boundary blow-up solutions for quasi-linear elliptic problems I: Large and small solutions, Proc Roy Soc Edinburgh Sect A, 135, 615-642 (2005) · Zbl 1129.35381
[23] Guo, Z. M.; Webb, J. R L., Structure of boundary blow-up solutions for quasi-linear elliptic problems II: Small and intermediate solutions, J Differential Equations, 211, 187-217 (2005) · Zbl 1134.35339
[24] Huang, S. B., Asymptotic behavior of boundary blow-up solutions to elliptic equations, Z Angew Math Phys, 67, 1-20 (2016) · Zbl 1339.35114
[25] Huang, S. B.; Li, W. T.; Tian, Q. Y., Large solution to nonlinear elliptic equation with nonlinear gradient terms, J Differential Equations, 251, 3297-3328 (2011) · Zbl 1231.35068
[26] Keller, J. B., On solutions of Δu = f(u), Comm Pure Appl Math, 10, 503-510 (1957) · Zbl 0090.31801
[27] Kondratév, V. A.; Nikishkin, V. A., Asymptotics near the boundary of a solution of a singular boundary-value problem for a semilinear elliptic equation, Differ Equ, 26, 345-348 (1990) · Zbl 0706.35054
[28] Lair, A. V., A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations, J Math Anal Appl, 240, 205-218 (1999) · Zbl 1058.35514
[29] Lair, A. V.; Wood, A. W., Large solutions of semilinear elliptic equations with nonlinear gradient terms, Int^J Math Sci, 22, 869-883 (1999) · Zbl 0951.35041
[30] Lasry, J. M.; Lions, P. L., Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints, 1: The model problem, Math Ann, 283, 583-630 (1989) · Zbl 0688.49026
[31] Lazer, A. C.; Mckenna, P. J., Asymptotic behavior of solutions of boundary blowup problems, Differential Integral Equations, 7, 1001-1019 (1994) · Zbl 0811.35010
[32] Leonori, T., Large solutions for a class of nonlinear elliptic equations with gradient terms, Adv Nonlinear Stud, 7, 237-269 (2007) · Zbl 1156.35030
[33] Leonori, T.; Porretta, A., The boundary behavior of blow-up solutions related to a stochastic control problem with state constraint, SIAM J Math Anal, 39, 1295-1327 (2007) · Zbl 1156.35009
[34] Lieberman, G. M., Asymptotic behavior and uniqueness of blow-up solutions of elliptic equations, Methods Appl Anal, 15, 243-262 (2008) · Zbl 1183.35141
[35] Loewner, C.; Nirenberg, L., Partial differential equations invariant under conformal or projective transformations (1974), New York: Academic Press, New York · Zbl 0298.35018
[36] López-Gómez, J., Metasolutions of Parabolic Equations in Population Dynamics (2016), New York: CRC Press, New York · Zbl 1344.35001
[37] Maric, V., Regular Variation and Differential Equations (2000), Lecture Notes in Mathematics, vol. 1726. Berlin: Springer-Verlag, Lecture Notes in Mathematics, vol. 1726. Berlin · Zbl 0946.34001
[38] Matero, J., Quasilinear elliptic equations with boundary blow-up, J Anal Math, 69, 229-247 (1996) · Zbl 0893.35032
[39] Mohammed, A., Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary value, J Math Anal Appl, 325, 480-489 (2007) · Zbl 1142.35412
[40] Osserman, R., On the inequality Δu > f(u), Pacific^J Math, 7, 1641-1647 (1957) · Zbl 0083.09402
[41] Porretta, A.; Véron, L., Asymptotic behaviour for the gradient of large solutions to some nonlinear elliptic equations, Adv Nonlinear Stud, 6, 351-378 (2006) · Zbl 1221.35141
[42] Resnick, S. I., Extreme Values, Regular Variation, and Point Processes (1987), New York-Berlin: Springer-Verlag, New York-Berlin · Zbl 0633.60001
[43] Seneta, R., Regular Varying Functions (1976), Lecture Notes in Mathematics, vol. 508. New York: Springer-Verlag, Lecture Notes in Mathematics, vol. 508. New York · Zbl 0324.26002
[44] Véron, L., Semilinear elliptic equations with uniform blowup on the boundary, J Anal Math, 59, 231-250 (1992) · Zbl 0802.35042
[45] Zeddini, N.; Alsaedi, R.; Mâagli, H., Exact boundary behavior of the unique positive solution to some singular elliptic problems, Nonlinear Anal, 89, 146-156 (2013) · Zbl 1281.31006
[46] Zhang, Z. J., Nonlinear elliptic equations with singular boundary conditions, J Math Anal Appl, 216, 390-397 (1997) · Zbl 0891.35041
[47] Zhang, Z. J., Boundary blow-up elliptic problems with nonlinear gradient terms, J Differential Equations, 228, 661-684 (2006) · Zbl 1130.35063
[48] Zhang, Z. J., Boundary blow-up elliptic problems with nonlinear gradient terms and singular weights, Proc Roy Soc Edinburgh Sect A, 138, 1403-1424 (2008) · Zbl 1155.35378
[49] Zhang, Z. J., Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms, Nonlinear Anal, 73, 3348-3363 (2010) · Zbl 1198.35077
[50] Zhang, Z. J., Large solutions of semilinear elliptic equations with a gradient term: Existence and boundary behavior, Commun Pure Appl Anal, 12, 1381-1392 (2013) · Zbl 1268.35041
[51] Zhang, Z. J., The existence and boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms, Adv Nonlinear Anal, 3, 165-185 (2014) · Zbl 1296.35067
[52] Zhang, Z. J., Boundary behavior of large solutions for semilinear elliptic equations with weights, Asymptot Anal, 96, 309-329 (2016) · Zbl 1354.35049
[53] Zhang, Z. J.; Ma, Y. J.; Mi, L., Blow-up rates of large solutions for elliptic equations, J Differential Equations, 249, 180-199 (2010) · Zbl 1191.35137
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