• Previous Article
    The sharp time decay rate of the isentropic Navier-Stokes system in $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $
  • ERA Home
  • This Issue
  • Next Article
    Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory
doi: 10.3934/era.2020119

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

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  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.

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, 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

[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. HuangW.-T. LiQ. 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. WanX. LiB. 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. HuangW.-T. LiQ. 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. WanX. LiB. 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

[1]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[2]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005

[3]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091

[4]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229

[5]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003

[6]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293

[7]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403

[8]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318

[9]

Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021007

[10]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

[11]

Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293

[12]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299

[13]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495

[14]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045

[15]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[16]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[17]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

[18]

Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021007

[19]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[20]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

 Impact Factor: 0.263

Metrics

  • PDF downloads (16)
  • HTML views (70)
  • Cited by (0)

Other articles
by authors

[Back to Top]