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A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior
School of Applied Science, Beijing Information Science & Technology University, Beijing 100192, China |
$ p $ |
$ \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*} $ |
$ \Delta_p u = \text{div}({|\nabla u|}^{p-2}\nabla u),\ 1<p<\infty $ |
$ \lambda_1 $ |
$ \lambda_2 $ |
$ \Omega $ |
$ \mathbb{R}^N,\ N\geq 2 $ |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Inc., Boston, MA, 1988.
![]() |
[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. |
[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. |
[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. |
[12] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc. Mineola, New York, 2006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
J. L. Vázquez,
A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Inc., Boston, MA, 1988.
![]() |
[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. |
[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. |
[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. |
[12] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc. Mineola, New York, 2006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
J. L. Vázquez,
A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[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. |
[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. |
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