Large solutions to elliptic systems of \(\infty \)-Laplacian equations. (English) Zbl 1468.35058
Summary: In this paper, we study the existence, uniqueness and boundary behavior of positive boundary blow-up solutions to the quasilinear system \(\Delta_{\infty}u=a(x)u^pv^q\), \(\Delta_{\infty}v=b(x)u^rv^s\) in a smooth bounded domain \(\Omega\subset\mathbb R^N\), with the explosive boundary condition \(u=v=+\infty\) on \(\partial\Omega \), where the operator \(\Delta_{\infty}\) is the \(\infty \)-Laplacian, the positive weight functions \(a(x)\), \(b(x)\) are Hölder continuous in \(\Omega \), and the exponents verify \(p, s > 3, q\), \(r>0\), and \((p-3)(s-3) > qr\).
MSC:
| 35J47 | Second-order elliptic systems |
| 35B44 | Blow-up in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
References:
| [1] | Juutinen, P.; Rossi, J. D., Large solutions for the infinity laplacian, Adv. Calc. Var., 1, 271-289 (2008) · Zbl 1158.35371 · doi:10.1515/ACV.2008.011 |
| [2] | Wang, W.; Gong, H. Z.; Zheng, S. N., Asymptotic estimates of boundary blow-up solutions to the infinity Laplace equations, J. Differential Equations., 256, 3, 3721-3742 (2014) · Zbl 1287.35038 · doi:10.1016/j.jde.2014.02.018 |
| [3] | Zhang, Z. J., Boundary behavior of large viscosity solutions to infinity Laplace equations, Z. Angew. Math. Phys., 66, 11, 1453-1472 (2015) · Zbl 1321.35049 · doi:10.1007/s00033-014-0470-1 |
| [4] | Mohammed, A.; Mohammed, S., On boundary blow-up solutions to equations involving the \(\infty \)-Laplacian, Nonlinear Anal., 74, 16, 5238-5252 (2011) · Zbl 1223.35141 · doi:10.1016/j.na.2011.04.022 |
| [5] | Lu, G. Z.; Wang, D. Y., Inhomogeneous infinity Laplacian equation, Adv. Math., 217, 1838-1868 (2008) · Zbl 1152.35042 · doi:10.1016/j.aim.2007.11.020 |
| [6] | Peres, Y.; Schramm, O.; Sheffield, S.; Wilson, D., Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22, 1, 167-210 (2009) · Zbl 1206.91002 · doi:10.1090/S0894-0347-08-00606-1 |
| [7] | Crandall, M. G.; Ishii, H.; Lions, P. L., User’s guide to viscosity solutions of sencond order partial differential equations, Bull. Amer. Math. Soc., 27, 1-67 (1992) · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5 |
| [8] | Cîrstea, F.; Râdulescu, V., Uniqueness of the blow-up Boundary solution of logistic equations with absorbtion, C. R. Acad. Sci. Paris., 335, 5, 447-452 (2002) · Zbl 1183.35124 · doi:10.1016/S1631-073X(02)02503-7 |
| [9] | Chen, Y. J., Boundary behavior for the large viscosity solutions to equations involving the infinity-Laplacian, Appl. Anal., 96, 12, 2065-2074 (2017) · Zbl 1380.35077 · doi:10.1080/00036811.2016.1202405 |
| [10] | Dancer, N.; Du, Y., Effects of certain degeneracies in the predator-prey model, J. Math. Anal, 34, 2, 292-314 (2002) · Zbl 1055.35046 |
| [11] | Melián, J. G.; Rossi, J. D., Boundary blow-up solutions to elliptic systems of competive type, J. Differential Equations, 206, 1, 156-181 (2004) · Zbl 1162.35359 · doi:10.1016/j.jde.2003.12.004 |
| [12] | Melián, J. G., A remark on uniqueness of large solutions of elliptic systems of competitive type, J. Math. Anal. Appl., 33, 1, 608-616 (2007) · Zbl 1131.35015 · doi:10.1016/j.jmaa.2006.09.006 |
| [13] | Mu, C.; Huang, S., Large solutions for an elliptic system of competitive type: existence, uniqueness, and asymptotic behavior, Nonlinear Anal., 71, 10, 4544-4552 (2009) · Zbl 1167.35373 · doi:10.1016/j.na.2009.03.012 |
| [14] | Melián, J. G., Large solutions for an elliptic system of quasilinear equations, J. Differential Equations, 245, 12, 3735-3752 (2008) · Zbl 1169.35021 · doi:10.1016/j.jde.2008.04.004 |
| [15] | Wang, Y.; Wang, M., Boundary blow up solutions for a cooperative system of quasilinear equations, J. Math. Anal. Appl., 368, 12, 736-744 (2010) · Zbl 1194.35141 · doi:10.1016/j.jmaa.2010.03.009 |
| [16] | Chen, Y. J.; Zhu, Y. P., Large solutions for a cooperative elliptic system of P-Laplacian equations, Nonlinear Anal., 73, 2, 450-457 (2010) · Zbl 1191.35119 · doi:10.1016/j.na.2010.03.036 |
| [17] | Wei, L.; Yang, Z. D., Large solutions of quasilinear elliptic system of competitive type: existence and asympeotic behavior, Int. J. Differ. Equ., 2010, 1-17 (2010) · Zbl 1207.35125 |
| [18] | Wei, L.; Cheng, X. Y.; Feng, Z. S., Existence and behavior of positive solutions to elliptic system with Hardy potential, Electron, J. Differential Equations., 2016, 223, 1-14 (2016) · Zbl 1352.35028 |
| [19] | Gómez, J. L.; Maire, L., Boundary blow-up rate and uniqueness of the large solution for an elliptic cooperative system of logistic type, Nonlinear Anal. Real World Appl., 33, 298-316 (2017) · Zbl 1352.35040 · doi:10.1016/j.nonrwa.2016.07.001 |
| [20] | Ladyzhenskaya, O. A.; Ural’tseva, N. N., Linear and Quasilinear Elliptic Equations, Acad. Press., 0, 0 (1968) · Zbl 0164.13002 |
| [21] | Benedetto, E. D., \(C^{1+\alpha}\) local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7, 8, 827-850 (1983) · Zbl 0539.35027 · doi:10.1016/0362-546X(83)90061-5 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
