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Blow-up rate of the unique solution for a class of one-dimensional \(p\)-Laplacian equations. (English) Zbl 1266.34053

The exact blow-up rate of the unique solution of the singular boundary value problem \[ \begin{aligned} (|u'|^{p-2}u')'=b(t)f(u),\;u(t)>0,& \quad t>0, \\ u(0)=\infty, \quad u(\infty)=0,& \end{aligned} \] is obtained, where \(1<p \leq 2\), \(b \in C^1(0,\infty)\) is positive and nondecreasing on \((0,\infty)\), \(f \in C^1[0,\infty)\), \(f(0)=0\), \(f(u)/u\) is increasing on \((0,\infty)\).

MSC:

34C11 Growth and boundedness of solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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[1] Keller, J. B., On solutions of \(\Delta u = f(u)\), Comm. Pure Appl. Math., 10, 503-510 (1957) · Zbl 0090.31801
[2] Osserman, R., On the inequality \(\Delta u \geq f(u)\), Pacific J. Math., 7, 1641-1647 (1957) · Zbl 0083.09402
[3] 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
[4] Cîrstea, F.; Raˇdulescu, V., Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Acad. Sci. Paris, Sér. I, 335, 447-452 (2002) · Zbl 1183.35124
[5] Cîrstea, F.; Raˇdulescu, V., Blow-up solutions for semilinear elliptic problems, Nonlinear Anal., 48, 541-554 (2002)
[6] Cîrstea, F.; Raˇdulescu, V., Asymptotics for the blow-up boundary solution of the logistic equation with absorption, C. R. Acad. Sci. Paris, Sér. I, 336, 231-236 (2003) · Zbl 1068.35035
[7] Cîrstea, F.; Raˇdulescu, V., Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach, Asymptot. Anal., 46, 275-298 (2006) · Zbl 1245.35037
[8] Cîrstea, F.; Raˇdulescu, V., Boundary blow-up in nonlinear elliptic equations of Bieberbach-Rademacher type, Trans. Amer. Math. Soc., 359, 3275-3286 (2007) · Zbl 1134.35039
[9] Cîrstea, F., Elliptic equations with competing rapidly varying nonlinearities and boundary blow-up, Adv. Differential Equations, 9, 995-1030 (2007) · Zbl 1162.35036
[10] Du, Y., (Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Maximum Principles and Applications, Series in Partial Differential Equations and Applications, 2, vol. 1 (2006), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ) · Zbl 1202.35043
[11] Du, Y.; Huang, Q., Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal., 31, 1-18 (1999) · Zbl 0959.35065
[12] García-Melián, J., Boundary behavior of large solutions to elliptic equations with singular weights, Nonlinear Anal., 67, 818-826 (2007) · Zbl 1143.35054
[13] García-Melián, J., Uniqueness of positive solutions for a boundary blow-up problem, J. Math. Anal. Appl., 360, 530-536 (2009) · Zbl 1182.35006
[14] Cano-Casanova, S.; López-Gómez, J., Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimension problems on the half-line, J. Differential Equations, 244, 3180-3203 (2008) · Zbl 1149.34020
[15] Zhang, Z.; Mi, L.; Yin, X., Blow-up rate of the unique solution for a class of one-dimensional problems on the half-line, J. Math. Anal. Appl., 348, 797-805 (2008) · Zbl 1176.34033
[16] Wei, L.; Zhu, J., The existence and blow-up rate of large solutions of one-dimensional \(p\)-Laplacian equations, Nonlinear Anal.: Real World Appl., 13, 665-676 (2012) · Zbl 1253.34035
[17] Seneta, R., (Regular Varying Functions. Regular Varying Functions, Lecture Notes in Mathematics, vol. 508 (1976), Springer-Verlag)
[18] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., (Regular Variation. Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27 (1987), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0617.26001
[19] Resnick, S. I., Extreme Values, Regular Variation, and Point Processes (1987), Springer-Verlag: Springer-Verlag New York, Berlin · Zbl 0633.60001
[20] Maric, V., (Regular Variation and Differential Equations. Regular Variation and Differential Equations, Lecture Notes in Mathematics, vol. 1726 (2000), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0946.34001
[21] de Haan, L., (On Regular Variation and its Application to the weak Convergence of Sample Extremes. On Regular Variation and its Application to the weak Convergence of Sample Extremes, Mathematics Centre Tract, vol. 32 (1970), University of Amsterdam: University of Amsterdam Amsterdam) · Zbl 0226.60039
[22] J.L. Geluk, L. de Haan, Regular variation, Extensions and Tauberian Theorems, CWI Tract, Centrum Wisk. Inform., Amsterdam, 1987.; J.L. Geluk, L. de Haan, Regular variation, Extensions and Tauberian Theorems, CWI Tract, Centrum Wisk. Inform., Amsterdam, 1987.
[23] Mohammed, A., Boundary asymptotic and uniqueness of solutions to the \(p\)-Laplacian with infnite boundary values, J. Math. Anal. Appl., 325, 480-489 (2007) · Zbl 1142.35412
[24] Huang, S.; Tian, Q., Asymptotic behavior of large solutions to \(p\)-Laplacian of Bieberbach-Rademacher type, Nonlinear Anal., 71, 5773-5780 (2009) · Zbl 1176.35080
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