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

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

Yichen Zhang and  Meiqiang Feng ,

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

* Corresponding author: Meiqiang Feng

Received  April 2020 Revised  June 2020 Published  December 2020 Early access  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 $
.
Keywords: p-Laplacian equatios, positive solutions, uniqueness and asymptotic behavior, eigenvalue theory in cones, Gustafson and Schmitt's fixed-point theorem.
Mathematics Subject Classification: 35J60, 35J66, 35J92.
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. Agarwal, H. 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. Castro, L. 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. Chu, D. 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. Liu, J. 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. Perera, R. 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. Shivaji, I. 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. Xiang, B. 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. Agarwal, H. 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. Castro, L. 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. Chu, D. 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. Liu, J. 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. Perera, R. 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. Shivaji, I. 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. Xiang, B. 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

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