December  2020, 28(4): 1419-1438. doi: 10.3934/era.2020075

A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior

School of Applied Science, Beijing Information Science & Technology University, Beijing 100192, China

* Corresponding author: Meiqiang Feng

Received  April 2020 Revised  June 2020 Published  July 2020

Fund Project: This work is sponsored by the National Natural Science Foundation of China (11301178) and the Beijing Natural Science Foundation of China (1163007)

We prove uniqueness, existence and asymptotic behavior of positive solutions to the system coupled by
$ p $
-Laplacian elliptic equations
$ \begin{align*} \left \{ \begin{array}{l} -\Delta_p z_1 = \lambda_1 g_1(z_2)\ \ {\rm in}\ \Omega,\\ -\Delta_p z_2 = \lambda_2 g_2(z_1)\ \ {\rm in}\ \Omega,\\ z_1 = z_2 = 0\ \ {\rm on}\ \ \partial \Omega, \end{array} \right. \end{align*} $
where
$ \Delta_p u = \text{div}({|\nabla u|}^{p-2}\nabla u),\ 1<p<\infty $
,
$ \lambda_1 $
and
$ \lambda_2 $
are positive parameters,
$ \Omega $
is the open unit ball in
$ \mathbb{R}^N,\ N\geq 2 $
.
Citation: 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
References:
[1]

R. P. AgarwalH. Lü and D. O'Regan, Eigenvalues and the one-dimensional $p$-Laplacian, J. Math. Anal. Appl., 266 (2002), 383-400.  doi: 10.1006/jmaa.2001.7742.  Google Scholar

[2]

A. CastroL. Sankar and R. Shivaji, Uniqueness of non-negative solutions for semipositone problems on exterior domains, J. Math. Anal. Appl., 394 (2012), 432-437.  doi: 10.1016/j.jmaa.2012.04.005.  Google Scholar

[3]

K. D. ChuD. D. Hai and R. Shivaji, Uniqueness of positive radial solutions for infinite semipositone $p$-Laplacian problems in exterior domains, J. Math. Anal. Appl., 472 (2019), 510-525.  doi: 10.1016/j.jmaa.2018.11.037.  Google Scholar

[4]

L. D'Ambrosio and E. Mitidieri, Quasilinear elliptic systems in divergence form associated to general nonlinearities, Adv. Nonlinear Anal., 7 (2018), 425-447.  doi: 10.1515/anona-2018-0171.  Google Scholar

[5]

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

[6]

Z. M. Guo, Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations, Appl. Anal., 47 (1992), 173-189.  doi: 10.1080/00036819208840139.  Google Scholar

[7]

Z. M. Guo and J. R. L. Webb, Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 189-198.  doi: 10.1017/S0308210500029280.  Google Scholar

[8] D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Inc., Boston, MA, 1988.   Google Scholar
[9]

G. B. Gustafson and K. Schmitt, Nonzero solutions of boundary value problems for second order ordinary and delay differential equations, J. Differential Equations, 12 (1972), 129-147.  doi: 10.1016/0022-0396(72)90009-5.  Google Scholar

[10]

D. D. Hai, Uniqueness of positive solutions for a class of semilinear elliptic systems, Nonlinear Anal., 52 (2003), 595-603.  doi: 10.1016/S0362-546X(02)00125-6.  Google Scholar

[11]

D. D. Hai and R. Shivaji, Existence and uniqueness for a class of quasilinear elliptic boundary value problems, J. Differential Equations, 193 (2003), 500-510.  doi: 10.1016/S0022-0396(03)00028-7.  Google Scholar

[12]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc. Mineola, New York, 2006.  Google Scholar

[13]

S.-S. Lin, On the number of positive solutions for nonlinear elliptic equations when a parameter is large, Nonlinear Anal., 16 (1991), 283-297.  doi: 10.1016/0362-546X(91)90229-T.  Google Scholar

[14]

P. Lindqvist, Notes on the $p$-Laplace Equation, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 102, University of Jyväskylä, Jyväskylä, 2006.  Google Scholar

[15]

Z. LiuJ. Su and Z.-Q. Wang, A twist condition and periodic solutions of Hamiltonian systems, Adv. Math., 218 (2008), 1895-1913.  doi: 10.1016/j.aim.2008.03.024.  Google Scholar

[16]

Z. Lou, T. Weth and Z. Zhang, Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type, Z. Angew. Math. Phys., 70 (2019), Paper No. 35, 19 pp. doi: 10.1007/s00033-019-1080-8.  Google Scholar

[17]

N. Mavinga and R. Pardo, A priori bounds and existence of positive solutions for semilinear elliptic systems, J. Math. Anal. Appl., 449 (2017), 1172-1188.  doi: 10.1016/j.jmaa.2016.12.058.  Google Scholar

[18]

K. PereraR. Shivaji and I. Sim, A class of semipositone $p$-Laplacian problems with a critical growth reaction term, Adv. Nonlinear Anal., 9 (2020), 516-525.  doi: 10.1515/anona-2020-0012.  Google Scholar

[19]

J. Sánchez, Multiple positive solutions of singular eigenvalue type problems involving the one-dimensional $p$-Laplacian, J. Math. Anal. Appl., 292 (2004), 401-414.  doi: 10.1016/j.jmaa.2003.12.005.  Google Scholar

[20]

R. ShivajiI. Sim and B. Son, A uniqueness result for a semipositone $p$-Laplacian problem on the exterior of a ball, J. Math. Anal. Appl., 445 (2017), 459-475.  doi: 10.1016/j.jmaa.2016.07.029.  Google Scholar

[21]

B. Son and P. Wang, Analysis of positive radial solutions for singular superlinear $p$-Laplacian systems on the exterior of a ball, Nonlinear Anal., 192 (2020), 111657, 15 pp. doi: 10.1016/j.na.2019.111657.  Google Scholar

[22]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.  Google Scholar

[23]

M. XiangB. Zhang and V. D. R$\breve{a}$dulescu, Existence of solutions for perturbed fractional $p$-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413.  doi: 10.1016/j.jde.2015.09.028.  Google Scholar

[24]

Z. Zhang and S. Li, On sign-changing andmultiple solutions of the $p$-Laplacian, J. Funct. Anal., 197 (2003), 447-468.  doi: 10.1016/S0022-1236(02)00103-9.  Google Scholar

show all references

References:
[1]

R. P. AgarwalH. Lü and D. O'Regan, Eigenvalues and the one-dimensional $p$-Laplacian, J. Math. Anal. Appl., 266 (2002), 383-400.  doi: 10.1006/jmaa.2001.7742.  Google Scholar

[2]

A. CastroL. Sankar and R. Shivaji, Uniqueness of non-negative solutions for semipositone problems on exterior domains, J. Math. Anal. Appl., 394 (2012), 432-437.  doi: 10.1016/j.jmaa.2012.04.005.  Google Scholar

[3]

K. D. ChuD. D. Hai and R. Shivaji, Uniqueness of positive radial solutions for infinite semipositone $p$-Laplacian problems in exterior domains, J. Math. Anal. Appl., 472 (2019), 510-525.  doi: 10.1016/j.jmaa.2018.11.037.  Google Scholar

[4]

L. D'Ambrosio and E. Mitidieri, Quasilinear elliptic systems in divergence form associated to general nonlinearities, Adv. Nonlinear Anal., 7 (2018), 425-447.  doi: 10.1515/anona-2018-0171.  Google Scholar

[5]

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

[6]

Z. M. Guo, Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations, Appl. Anal., 47 (1992), 173-189.  doi: 10.1080/00036819208840139.  Google Scholar

[7]

Z. M. Guo and J. R. L. Webb, Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 189-198.  doi: 10.1017/S0308210500029280.  Google Scholar

[8] D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Inc., Boston, MA, 1988.   Google Scholar
[9]

G. B. Gustafson and K. Schmitt, Nonzero solutions of boundary value problems for second order ordinary and delay differential equations, J. Differential Equations, 12 (1972), 129-147.  doi: 10.1016/0022-0396(72)90009-5.  Google Scholar

[10]

D. D. Hai, Uniqueness of positive solutions for a class of semilinear elliptic systems, Nonlinear Anal., 52 (2003), 595-603.  doi: 10.1016/S0362-546X(02)00125-6.  Google Scholar

[11]

D. D. Hai and R. Shivaji, Existence and uniqueness for a class of quasilinear elliptic boundary value problems, J. Differential Equations, 193 (2003), 500-510.  doi: 10.1016/S0022-0396(03)00028-7.  Google Scholar

[12]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc. Mineola, New York, 2006.  Google Scholar

[13]

S.-S. Lin, On the number of positive solutions for nonlinear elliptic equations when a parameter is large, Nonlinear Anal., 16 (1991), 283-297.  doi: 10.1016/0362-546X(91)90229-T.  Google Scholar

[14]

P. Lindqvist, Notes on the $p$-Laplace Equation, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 102, University of Jyväskylä, Jyväskylä, 2006.  Google Scholar

[15]

Z. LiuJ. Su and Z.-Q. Wang, A twist condition and periodic solutions of Hamiltonian systems, Adv. Math., 218 (2008), 1895-1913.  doi: 10.1016/j.aim.2008.03.024.  Google Scholar

[16]

Z. Lou, T. Weth and Z. Zhang, Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type, Z. Angew. Math. Phys., 70 (2019), Paper No. 35, 19 pp. doi: 10.1007/s00033-019-1080-8.  Google Scholar

[17]

N. Mavinga and R. Pardo, A priori bounds and existence of positive solutions for semilinear elliptic systems, J. Math. Anal. Appl., 449 (2017), 1172-1188.  doi: 10.1016/j.jmaa.2016.12.058.  Google Scholar

[18]

K. PereraR. Shivaji and I. Sim, A class of semipositone $p$-Laplacian problems with a critical growth reaction term, Adv. Nonlinear Anal., 9 (2020), 516-525.  doi: 10.1515/anona-2020-0012.  Google Scholar

[19]

J. Sánchez, Multiple positive solutions of singular eigenvalue type problems involving the one-dimensional $p$-Laplacian, J. Math. Anal. Appl., 292 (2004), 401-414.  doi: 10.1016/j.jmaa.2003.12.005.  Google Scholar

[20]

R. ShivajiI. Sim and B. Son, A uniqueness result for a semipositone $p$-Laplacian problem on the exterior of a ball, J. Math. Anal. Appl., 445 (2017), 459-475.  doi: 10.1016/j.jmaa.2016.07.029.  Google Scholar

[21]

B. Son and P. Wang, Analysis of positive radial solutions for singular superlinear $p$-Laplacian systems on the exterior of a ball, Nonlinear Anal., 192 (2020), 111657, 15 pp. doi: 10.1016/j.na.2019.111657.  Google Scholar

[22]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.  Google Scholar

[23]

M. XiangB. Zhang and V. D. R$\breve{a}$dulescu, Existence of solutions for perturbed fractional $p$-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413.  doi: 10.1016/j.jde.2015.09.028.  Google Scholar

[24]

Z. Zhang and S. Li, On sign-changing andmultiple solutions of the $p$-Laplacian, J. Funct. Anal., 197 (2003), 447-468.  doi: 10.1016/S0022-1236(02)00103-9.  Google Scholar

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054

[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]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020404

[11]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[12]

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

[13]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

[14]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[15]

Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020178

[16]

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

[17]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294

[18]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[19]

Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020123

[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 (79)
  • HTML views (204)
  • Cited by (0)

Other articles
by authors

[Back to Top]