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[J]. Electronic Research Archive, 2020, 28(4): 1419-1438. doi: 10.3934/era.2020075
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 $.
[1] |
Eigenvalues and the one-dimensional $p$-Laplacian. J. Math. Anal. Appl. (2002) 266: 383-400. ![]() |
[2] |
Uniqueness of non-negative solutions for semipositone problems on exterior domains. J. Math. Anal. Appl. (2012) 394: 432-437. ![]() |
[3] |
Uniqueness of positive radial solutions for infinite semipositone $p$-Laplacian problems in exterior domains. J. Math. Anal. Appl. (2019) 472: 510-525. ![]() |
[4] |
Quasilinear elliptic systems in divergence form associated to general nonlinearities. Adv. Nonlinear Anal. (2018) 7: 425-447. ![]() |
[5] |
Boundary blow-up solutions and the applications in quasilinear elliptic equations. J. Anal. Math. (2003) 89: 277-302. ![]() |
[6] |
Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations. Appl. Anal. (1992) 47: 173-189. ![]() |
[7] |
Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large. Proc. Roy. Soc. Edinburgh Sect. A (1994) 124: 189-198. ![]() |
[8] | (1988) Nonlinear Problems in Abstract Cones.Academic Press. |
[9] |
Nonzero solutions of boundary value problems for second order ordinary and delay differential equations. J. Differential Equations (1972) 12: 129-147. ![]() |
[10] |
Uniqueness of positive solutions for a class of semilinear elliptic systems. Nonlinear Anal. (2003) 52: 595-603. ![]() |
[11] |
Existence and uniqueness for a class of quasilinear elliptic boundary value problems. J. Differential Equations (2003) 193: 500-510. ![]() |
[12] | J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc. Mineola, New York, 2006. |
[13] |
On the number of positive solutions for nonlinear elliptic equations when a parameter is large. Nonlinear Anal. (1991) 16: 283-297. ![]() |
[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] |
A twist condition and periodic solutions of Hamiltonian systems. Adv. Math. (2008) 218: 1895-1913. ![]() |
[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] |
A priori bounds and existence of positive solutions for semilinear elliptic systems. J. Math. Anal. Appl. (2017) 449: 1172-1188. ![]() |
[18] |
A class of semipositone $p$-Laplacian problems with a critical growth reaction term. Adv. Nonlinear Anal. (2020) 9: 516-525. ![]() |
[19] |
Multiple positive solutions of singular eigenvalue type problems involving the one-dimensional $p$-Laplacian. J. Math. Anal. Appl. (2004) 292: 401-414. ![]() |
[20] |
A uniqueness result for a semipositone $p$-Laplacian problem on the exterior of a ball. J. Math. Anal. Appl. (2017) 445: 459-475. ![]() |
[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] |
A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. (1984) 12: 191-202. ![]() |
[23] |
Existence of solutions for perturbed fractional $p$-Laplacian equations. J. Differential Equations (2016) 260: 1392-1413. ![]() |
[24] |
On sign-changing andmultiple solutions of the $p$-Laplacian. J. Funct. Anal. (2003) 197: 447-468. ![]() |