Figure 1.
Bifurcation of solution from the origin.
-
Previous Article
Cyclicity of degenerate graphic $DF_{2a}$ of Dumortier-Roussarie-Rousseau program
- CPAA Home
- This Issue
-
Next Article
Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application
May
2018, 17(3): 1295-1304.
doi: 10.3934/cpaa.2018062
Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system
| 1. | Department of Mathematics, Indian Institute of Technology Madras, Chennai-600036, India |
| 2. | Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, USA |
| 3. | Department of Mathematics, Wayne State University, Detroit, MI 48202, USA |
In this paper we study the positive solutions to the
| $n\times n$ |
| $p$ |
| $\begin{equation*}\begin{cases}-\left(\varphi_{p_1}(u_1')\right)' = \lambda h_1(t) \left(u_1^{p_1-1-\alpha_1}+f_1(u_2)\right),\quad t\in (0,1),\\-\left(\varphi_{p_2}(u_2')\right)' = \lambda h_2(t) \left(u_2^{p_2-1-\alpha_2}+f_2(u_3)\right),\quad t\in (0,1),\\\quad\quad\quad\vdots\qquad\,\: =\quad\quad\quad\quad\quad\quad \vdots\\-\left(\varphi_{p_n}(u_n')\right)' = \lambda h_n(t) \left(u_n^{p_n-1-\alpha_n}+f_n(u_1)\right),~~\, t\in (0,1),\\\quad\,\,\,\, u_j(0)=0=u_j(1); ~~ j=1,2,\dots,n, \\ \end{cases}\end{equation*}$ |
where
| $\lambda$ |
| $p_j>1$ |
| $\alpha_j\in(0,p_j-1)$ |
| $\varphi_{p_j}(w)=|w|^{p_j-2}w$ |
| $h_j \in C((0,1),(0, \infty))\cap L^1((0,1),(0,\infty))$ |
| $j=1,2,\dots,n$ |
| $f_j:[0,\infty)\rightarrow[0,\infty)$ |
| $j=1,2,\dots,n$ |
| $f_j(0)=0$ |
| $\lambda>0$ |
| $\lambda$ |
Mathematics Subject Classification:
Primary: 34B16, 34B18; Secondary: 35J57.
Citation:
Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure & Applied Analysis,
2018, 17
(3)
: 1295-1304.
doi: 10.3934/cpaa.2018062
![]()
References:
| [1] |
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 229-256. Google Scholar |
| [2] |
Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional p-Laplacian, J. Differential Equations, 229 (2006), 620-709. Google Scholar |
| [3] |
R. Manásevich and J. Mawhin, Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators, J. Korean Math. Soc., 37 (2000), 665-685. Google Scholar |
| [4] |
R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Applications, Lecture Notes in Pure and Appl. Math., 109 (1987), 561-566. Google Scholar |
| [5] |
R. Shivaji and B. Son, Bifurcation and multiplicity results for classes of $p,q$ Laplacian systems, Topol. Methods Nonlinear Anal., 48 (2016), 103-114. Google Scholar |
show all references
References:
| [1] |
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 229-256. Google Scholar |
| [2] |
Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional p-Laplacian, J. Differential Equations, 229 (2006), 620-709. Google Scholar |
| [3] |
R. Manásevich and J. Mawhin, Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators, J. Korean Math. Soc., 37 (2000), 665-685. Google Scholar |
| [4] |
R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Applications, Lecture Notes in Pure and Appl. Math., 109 (1987), 561-566. Google Scholar |
| [5] |
R. Shivaji and B. Son, Bifurcation and multiplicity results for classes of $p,q$ Laplacian systems, Topol. Methods Nonlinear Anal., 48 (2016), 103-114. Google Scholar |
Figure 2.
Bifurcation for all $\lambda>0.$
Figure 3.
Multiplicity results for certain range of $\lambda$ .
| [1] |
K. D. Chu, D. D. Hai. Positive solutions for the one-dimensional singular superlinear $ p $-Laplacian problem. Communications on Pure & Applied Analysis, 2020, 19 (1) : 241-252. doi: 10.3934/cpaa.2020013 |
| [2] |
Said Taarabti. Positive solutions for the $ p(x)- $Laplacian : Application of the Nehari method. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021029 |
| [3] |
Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026 |
| [4] |
Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3497-3528. doi: 10.3934/dcdss.2020442 |
| [5] |
Claudianor O. Alves, Vincenzo Ambrosio, Teresa Isernia. Existence, multiplicity and concentration for a class of fractional $ p \& q $ Laplacian problems in $ \mathbb{R} ^{N} $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2009-2045. doi: 10.3934/cpaa.2019091 |
| [6] |
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 |
| [7] |
Jiayi Han, Changchun Liu. Global existence for a two-species chemotaxis-Navier-Stokes system with $ p $-Laplacian. Electronic Research Archive, , () : -. doi: 10.3934/era.2021050 |
| [8] |
Mihai Mihăilescu, Julio D. Rossi. Monotonicity with respect to $ p $ of the First Nontrivial Eigenvalue of the $ p $-Laplacian with Homogeneous Neumann Boundary Conditions. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4363-4371. doi: 10.3934/cpaa.2020198 |
| [9] |
Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian. Mathematical Control & Related Fields, 2020, 10 (4) : 827-854. doi: 10.3934/mcrf.2020021 |
| [10] |
Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2635-3652. doi: 10.3934/dcds.2020378 |
| [11] |
Ziqing Yuan, Jianshe Yu. Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3285-3303. doi: 10.3934/dcdss.2020281 |
| [12] |
Anhui Gu. Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3863-3878. doi: 10.3934/dcdsb.2020266 |
| [13] |
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, 2021, 41 (7) : 3211-3240. doi: 10.3934/dcds.2020403 |
| [14] |
Junyong Eom, Ryuichi Sato. Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4373-4386. doi: 10.3934/cpaa.2020199 |
| [15] |
Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293 |
| [16] |
Changchun Liu, Pingping Li. Global existence for a chemotaxis-haptotaxis model with $ p $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1399-1419. doi: 10.3934/cpaa.2020070 |
| [17] |
Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control & Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030 |
| [18] |
Lujuan Yu. The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2621-2637. doi: 10.3934/dcdsb.2020025 |
| [19] |
Xin-Guang Yang, Marcelo J. D. Nascimento, Maurício L. Pelicer. Uniform attractors for non-autonomous plate equations with $ p $-Laplacian perturbation and critical nonlinearities. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1937-1961. doi: 10.3934/dcds.2020100 |
| [20] |
Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]