Asymptotic estimates of boundary blow-up solutions to the infinity Laplace equations. (English) Zbl 1287.35038
Summary: In this paper we study the asymptotic behavior of boundary blow-up solutions to the equation \({\Delta_\infty}u=b(x)f(u)\) in \({\varOmega}\), where \({\Delta}_\infty\) is the \(\infty\)-Laplacian, the nonlinearity \( f\) is a positive, increasing function in (\(0,\infty\)), and the weighted function \(b\in C({\overline\varOmega})\) is positive in \({\varOmega}\) and may vanish on the boundary. We first establish the exact boundary blow-up estimates with the first expansion when f is regularly varying at infinity with index \(p>3\) and the weighted function b is controlled on the boundary in some manner. Furthermore, for the case of \(f(s)=s^p(1+\tilde cg(s))\), with the function \( g\) normalized regularly varying with index \(-q<0\), we obtain the second expansion of solutions near the boundary. It is interesting that the second term in the asymptotic expansion of boundary blow-up solutions to the infinity Laplace equation is independent of the geometry of the domain, quite different from the boundary blow-up problems involving the classical Laplacian.
MSC:
| 35J70 | Degenerate elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
Keywords:
infinity Laplacian; asymptotic estimate; boundary blow-up; first and second expansions; comparison principleReferences:
| [1] | Aronsson, G., Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6, 551-561 (1967) · Zbl 0158.05001 |
| [2] | Aronsson, G.; Crandall, M. G.; Juutinen, P., A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41, 439-505 (2004) · Zbl 1150.35047 |
| [3] | Anedda, C.; Porru, G., Second order estimates for boundary blow-up solutions of elliptic equations, Discrete Contin. Dyn. Syst., Suppl., 54-63 (2007) · Zbl 1163.35350 |
| [4] | Anedda, C.; Porru, G., Boundary behavior for solutions of boundary blow-up problems in a borderline case, J. Math. Anal. Appl., 352, 35-47 (2009) · Zbl 1168.35009 |
| [5] | Bandle, C., Asymptotic behavior of large solutions of quasilinear elliptic problems, Z. Angew. Math. Phys., 54, 731-738 (2003) · Zbl 1099.35505 |
| [6] | Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation, Encyclopedia Math. Appl., vol. 27 (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0617.26001 |
| [7] | 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 |
| [8] | Bandle, C.; Marcus, M., On second-order effects in the boundary behavior of large solutions of semilinear elliptic problems, Differential Integral Equations, 11, 23-34 (1998) · Zbl 1042.35535 |
| [9] | Bandle, C.; Marcus, M., Dependence of blow-up rate of large solutions of semilinear elliptic equations, on the curvature of the boundary, Complex Var., 49, 555-570 (2004) · Zbl 1129.35312 |
| [10] | Bhattacharya, T.; Mohammed, A., On solutions to Dirichlet problems involving the infinity-Laplacian, Adv. Calc. Var., 4, 445-487 (2011) · Zbl 1232.35056 |
| [11] | Bhattacharya, T.; Mohammed, A., Inhomogeneous Dirichlet problems involving the infinity-Laplacian, Adv. Differential Equations, 17, 225-266 (2012) · Zbl 1258.35094 |
| [12] | Crandall, M. G.; Evans, L. C.; Gariepy, R. F., Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations, 13, 123-139 (2001) · Zbl 0996.49019 |
| [13] | Crandall, M. G.; Lions, P. L., Viscosity solutions and Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277, 1-42 (1983) · Zbl 0599.35024 |
| [14] | Cano-Casanova, S.; López-Gómez, J., Blow-up rates of radially symmetric large solutions, J. Math. Anal. Appl., 352, 166-174 (2009) · Zbl 1163.35015 |
| [15] | Cîrstea, F.; Rǎdulescu, V., Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Acad. Sci. Paris, Ser. I, 335, 447-452 (2002) · Zbl 1183.35124 |
| [16] | Cîrstea, F.; Rǎdulescu, V., Asymptotics for the blow-up boundary solution of the logistic equation with absorption, C. R. Acad. Sci. Paris, Ser. I, 336, 231-236 (2003) · Zbl 1068.35035 |
| [17] | Cîrstea, F.; Rǎdulescu, V., Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach, Asymptot. Anal., 46, 275-298 (2006) · Zbl 1245.35037 |
| [18] | Du, Y.; Guo, Z., Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math., 89, 277-302 (2003) · Zbl 1162.35028 |
| [19] | 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 |
| [20] | Evans, L. C.; Savin, O., \(C^{1, \alpha}\) regularity for infinity harmonic functions in two dimensions, Calc. Var. Partial Differential Equations, 32, 325-347 (2008) · Zbl 1151.35096 |
| [21] | García-Melián, J.; Letelier-Albornoz, R.; Sabina de Lis, J., Uniqueness and asymptotic behavior for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. Soc., 129, 3593-3602 (2001) · Zbl 0989.35044 |
| [22] | García-Melián, J., Boundary behavior of large solutions to elliptic equations with singular weights, Nonlinear Anal., 67, 818-826 (2007) · Zbl 1143.35054 |
| [23] | Gladiali, F.; Porru, G., Estimates for explosive solutions to \(p\)-Laplace equations, (Progress in Partial Differential Equations, vol. 1. Progress in Partial Differential Equations, vol. 1, Pont-á-Mousson 1997. Progress in Partial Differential Equations, vol. 1. Progress in Partial Differential Equations, vol. 1, Pont-á-Mousson 1997, Pitman Res. Notes Math. Series, vol. 383 (1998), Longman), 117-127 · Zbl 0916.35041 |
| [24] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer Berlin · Zbl 0562.35001 |
| [25] | Jensen, R., Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal., 123, 51-74 (1993) · Zbl 0789.35008 |
| [26] | Juutinen, P., Principal eigenvalue of a very badly degenerate operator and applications, J. Differential Equations, 236, 532-550 (2007) · Zbl 1132.35066 |
| [27] | Keller, J. B., On solutions of \(\Delta u = f(u)\), Comm. Pure Appl. Math., 10, 503-510 (1957) · Zbl 0090.31801 |
| [28] | López-Gómez, J., Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations, 224, 385-439 (2006) · Zbl 1208.35036 |
| [29] | López-Gómez, J., Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst., Suppl., 677-686 (2007) · Zbl 1163.35352 |
| [30] | Lazer, A. C.; McKenna, P. J., Asymptotic behavior of solutions of boundary blow-up problems, Differential Integral Equations, 7, 1001-1019 (1994) · Zbl 0811.35010 |
| [31] | Lindqvist, P.; Manfredi, J., The Harnack inequality for ∞-harmonic functions, Electron. J. Differential Equations, 1995, 1-5 (1995) · Zbl 0818.35033 |
| [32] | Lu, G. Z.; Wang, P. Y., Inhomogeneous infinity Laplace equation, Adv. Math., 217, 1838-1868 (2008) · Zbl 1152.35042 |
| [33] | Lu, G. Z.; Wang, P. Y., A PDE perspective of the normalized infinity Laplacian, Comm. Partial Differential Equations, 10, 1788-1817 (2008) · Zbl 1157.35388 |
| [34] | Lu, G. Z.; Wang, P. Y., A uniqueness theorem for degenerate elliptic equations, (Geometric Methods in PDE’s. Geometric Methods in PDE’s, Lect. Notes Semin. Interdiscip. Mat., vol. 7 (2008), Semin. Interdiscip. Mat. (S.I.M.): Semin. Interdiscip. Mat. (S.I.M.) Potenza), 207-222 · Zbl 1187.35109 |
| [35] | Lu, G. Z.; Wang, P. Y., Infinity Laplace equation with non-trivial right-hand side, Electron. J. Differential Equations, 2010, 1-12 (2010) · Zbl 1194.35194 |
| [36] | Mohammed, A., Boundary asymptotic and uniqueness of solutions to the \(p\)-Laplacian with infinite boundary values, J. Math. Anal. Appl., 325, 480-489 (2007) · Zbl 1142.35412 |
| [37] | Mohammed, A.; Mohammed, S., On boundary blow-up solutions to equations involving the ∞-Laplacian, Nonlinear Anal., 74, 5238-5252 (2011) · Zbl 1223.35141 |
| [38] | Osserman, R., On the inequality \(\Delta u \geqslant f(u)\), Pacific J. Math., 7, 1641-1647 (1957) · Zbl 0083.09402 |
| [39] | Peres, Y.; Schramm, O.; Sheffield, S.; Wilson, D., Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22, 167-210 (2009) · Zbl 1206.91002 |
| [40] | Rǎdulescu, V., Singular phenomena in nonlinear elliptic problems: from boundary blow-up solutions to equations with singular nonlinearities, (Chipot, Michel, Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 4 (2007)), 483-591 |
| [41] | Savin, O., \(C^1\) regularity for infinity harmonic functions in two dimensions, Arch. Ration. Mech. Anal., 176, 351-361 (2005) · Zbl 1112.35070 |
| [42] | Zhang, Z., The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems, J. Math. Anal. Appl., 308, 532-540 (2005) · Zbl 1160.35417 |
| [43] | Zhang, Z.; Ma, Y.; Mi, L.; Li, X., Blow-up rates of large solutions for elliptic equations, J. Differential Equations, 249, 180-199 (2010) · Zbl 1191.35137 |
| [44] | Zhang, Z., The second expansion of large solutions for semilinear elliptic equations, Nonlinear Anal., 74, 3445-3457 (2011) · Zbl 1217.35074 |
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