This paper is considered with the quasilinear elliptic equation $ \Delta_{p}u = b(x)f(u),\,u(x)>0,\,x\in\Omega, $ where $ \Omega $ is an exterior domain with compact smooth boundary, $ b\in \rm C(\Omega) $ is non-negative in $ \Omega $ and may be singular or vanish on $ \partial\Omega $, $ f\in C[0, \infty) $ is positive and increasing on $ (0, \infty) $ which satisfies a generalized Keller-Osserman condition and is regularly varying at infinity with critical index $ p-1 $. By structuring a new comparison function, we establish the new asymptotic behavior of large solutions to the above equation in the exterior domain. We find that the lower term of $ f $ has an important influence on the asymptotic behavior of large solutions. And then we further establish the uniqueness of such solutions.
Citation: Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case[J]. Electronic Research Archive, 2021, 29(3): 2359-2373. doi: 10.3934/era.2020119
This paper is considered with the quasilinear elliptic equation $ \Delta_{p}u = b(x)f(u),\,u(x)>0,\,x\in\Omega, $ where $ \Omega $ is an exterior domain with compact smooth boundary, $ b\in \rm C(\Omega) $ is non-negative in $ \Omega $ and may be singular or vanish on $ \partial\Omega $, $ f\in C[0, \infty) $ is positive and increasing on $ (0, \infty) $ which satisfies a generalized Keller-Osserman condition and is regularly varying at infinity with critical index $ p-1 $. By structuring a new comparison function, we establish the new asymptotic behavior of large solutions to the above equation in the exterior domain. We find that the lower term of $ f $ has an important influence on the asymptotic behavior of large solutions. And then we further establish the uniqueness of such solutions.
[1] | Uniqueness of the blow-up boundary solution of logistic equations with absorbtion. C. R. Math. Acad. Sci. Paris (2002) 335: 447-452. |
[2] | Large and entire large solution for a quasilinear problem. Nonlinear Anal. (2009) 70: 1738-1745. |
[3] | Explosive solutions of quasilinear elliptic equations: Existence and uniqueness. Nonlinear Anal. (1993) 20: 97-125. |
[4] | Boundary blow-up solutions and their applications in quasilinear elliptic equations. J. Anal. Math. (2003) 89: 277-302. |
[5] | D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 3$^rd$ ed., Springer-Verlag, Berlin, 1998. |
[6] | F. Gladiali and G. Porru, Estimates for Explosive Solutions to $p$-Laplace Equations, in: Progress in Partial Differential Equations, Pont-á-Mousson, 1997, vol. 1, in: Pitman Res. Notes Math. Ser., vol. 383, Longman, Harlow, 1998. |
[7] | S. Huang, Asymptotic behavior of boundary blow-up solutions to elliptic equations, Z. Angew. Math. Phys., 67 (2016), Art. 3, 20 pp. doi: 10.1007/s00033-015-0606-y |
[8] | General uniqueness results and blow-up rates for large solutions of elliptic equations. Proc. Roy. Soc. Edinburgh Sect. A (2012) 142: 825-837. |
[9] | Quasilinear elliptic equations with boundary blow-up. J. Anal. Math. (1996) 69: 229-247. |
[10] | Existence and asymptotic behavior of blow-up solutions to weighted quasilinear equations. J. Math. Anal. Appl. (2004) 298: 621-637. |
[11] | Boundary asymptotic and uniqueness of solutions to the $p$-Laplacian with infinite boundary values. J. Math. Anal. Appl. (2007) 325: 480-489. |
[12] | S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, Berlin, 1987. doi: 10.1007/978-0-387-75953-1 |
[13] | E. Seneta, Regular Varying Functions, in: Lecture Notes in Math., vol. 508, Springer-Verlag, 1976. doi: 10.1007/bfb0079659 |
[14] | H. Wan, Asymptotic behavior and uniqueness of entire large solutions to a quasilinear elliptic equation, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), Paper No. 30, 17 pp. doi: 10.14232/ejqtde.2017.1.30 |
[15] | Entire large solutions to semilinear elliptic equations with rapidly or regularly varying nonlinearities. Nonlinear Anal.: Real World Applications (2019) 45: 506-530. |
[16] | Existence of explosive positive solutions of quasilinear elliptic equations. Appl. Math. Comput. (2006) 177: 581-588. |
[17] | Boundary behavior of large solutions for semilinear elliptic equations with weights. Asymptot. Anal. (2016) 96: 309-329. |
[18] | Boundary behavior of large solutions to $p$-Laplacian elliptic equations. Nonlinear Anal.: Real World Applications (2017) 33: 40-57. |