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August  2021, 29(3): 2359-2373. doi: 10.3934/era.2020119

Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case

Yongxiu Shi and  Haitao Wan ,

School of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264005, China

* Corresponding author: Haitao Wan

Received  August 2020 Revised  October 2020 Published  August 2021 Early access  November 2020

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.

Keywords: p-Laplacian, large solutions, the exact asymptotic behavior, uniqueness, exterior domains.
Mathematics Subject Classification: Primary: 35J25, 35B40; Secondary: 35J62.
Citation: Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, 2021, 29 (3) : 2359-2373. doi: 10.3934/era.2020119
References:
[1]

F.-C. Cîrstea and V. Rǎdulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Math. Acad. Sci. Paris, 335 (2002), 447-452.  doi: 10.1016/S1631-073X(02)02503-7.  Google Scholar

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Y. Du and Z. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math., 89 (2003), 277-302.  doi: 10.1007/BF02893084.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 3$^rd$ ed., Springer-Verlag, Berlin, 1998.  Google Scholar

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J. Matero, Quasilinear elliptic equations with boundary blow-up, J. Anal. Math., 69 (1996), 229-247.  doi: 10.1007/BF02787108.  Google Scholar

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A. Mohammed, Boundary asymptotic and uniqueness of solutions to the $p$-Laplacian with infinite boundary values, J. Math. Anal. Appl., 325 (2007), 480-489.  doi: 10.1016/j.jmaa.2006.02.008.  Google Scholar

[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.  Google Scholar

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E. Seneta, Regular Varying Functions, in: Lecture Notes in Math., vol. 508, Springer-Verlag, 1976. doi: 10.1007/bfb0079659.  Google Scholar

[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.  Google Scholar

[15]

H. Wan, X. Li, B. Li and Y. Shi, Entire large solutions to semilinear elliptic equations with rapidly or regularly varying nonlinearities, Nonlinear Anal.: Real World Applications, 45 (2019), 506-530.  doi: 10.1016/j.nonrwa.2018.07.021.  Google Scholar

[16]

Z. Yang, Existence of explosive positive solutions of quasilinear elliptic equations, Appl. Math. Comput., 177 (2006), 581-588.  doi: 10.1016/j.amc.2005.09.088.  Google Scholar

[17]

Z. Zhang, Boundary behavior of large solutions for semilinear elliptic equations with weights, Asymptot. Anal., 96 (2016), 309-329.  doi: 10.3233/ASY-151345.  Google Scholar

[18]

Z. Zhang, Boundary behavior of large solutions to $p$-Laplacian elliptic equations, Nonlinear Anal.: Real World Applications, 33 (2017), 40-57.  doi: 10.1016/j.nonrwa.2016.05.008.  Google Scholar

show all references

References:
[1]

F.-C. Cîrstea and V. Rǎdulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Math. Acad. Sci. Paris, 335 (2002), 447-452.  doi: 10.1016/S1631-073X(02)02503-7.  Google Scholar

[2]

D.-P. Covei, Large and entire large solution for a quasilinear problem, Nonlinear Anal., 70 (2009), 1738-1745.  doi: 10.1016/j.na.2008.02.057.  Google Scholar

[3]

G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal., 20 (1993), 97-125.  doi: 10.1016/0362-546X(93)90012-H.  Google Scholar

[4]

Y. Du and Z. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math., 89 (2003), 277-302.  doi: 10.1007/BF02893084.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 3$^rd$ ed., Springer-Verlag, Berlin, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. Huang, W.-T. Li, Q. Tian and Y. Mi, General uniqueness results and blow-up rates for large solutions of elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 825-837.  doi: 10.1017/S0308210511000060.  Google Scholar

[9]

J. Matero, Quasilinear elliptic equations with boundary blow-up, J. Anal. Math., 69 (1996), 229-247.  doi: 10.1007/BF02787108.  Google Scholar

[10]

A. Mohammed, Existence and asymptotic behavior of blow-up solutions to weighted quasilinear equations, J. Math. Anal. Appl., 298 (2004), 621-637.  doi: 10.1016/j.jmaa.2004.05.030.  Google Scholar

[11]

A. Mohammed, Boundary asymptotic and uniqueness of solutions to the $p$-Laplacian with infinite boundary values, J. Math. Anal. Appl., 325 (2007), 480-489.  doi: 10.1016/j.jmaa.2006.02.008.  Google Scholar

[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.  Google Scholar

[13]

E. Seneta, Regular Varying Functions, in: Lecture Notes in Math., vol. 508, Springer-Verlag, 1976. doi: 10.1007/bfb0079659.  Google Scholar

[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.  Google Scholar

[15]

H. Wan, X. Li, B. Li and Y. Shi, Entire large solutions to semilinear elliptic equations with rapidly or regularly varying nonlinearities, Nonlinear Anal.: Real World Applications, 45 (2019), 506-530.  doi: 10.1016/j.nonrwa.2018.07.021.  Google Scholar

[16]

Z. Yang, Existence of explosive positive solutions of quasilinear elliptic equations, Appl. Math. Comput., 177 (2006), 581-588.  doi: 10.1016/j.amc.2005.09.088.  Google Scholar

[17]

Z. Zhang, Boundary behavior of large solutions for semilinear elliptic equations with weights, Asymptot. Anal., 96 (2016), 309-329.  doi: 10.3233/ASY-151345.  Google Scholar

[18]

Z. Zhang, Boundary behavior of large solutions to $p$-Laplacian elliptic equations, Nonlinear Anal.: Real World Applications, 33 (2017), 40-57.  doi: 10.1016/j.nonrwa.2016.05.008.  Google Scholar

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