Figure 1.
Bifurcation of solution from the origin.
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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 and Applied Analysis,
2018, 17
(3)
: 1295-1304.
doi: 10.3934/cpaa.2018062
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References:
| [1] |
H. Amann,
Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 229-256.
|
| [2] |
Y. H. Lee and I. Sim,
Global bifurcation phenomena for singular one-dimensional p-Laplacian, J. Differential Equations, 229 (2006), 620-709.
|
| [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.
|
| [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. |
| [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.
|
show all references
References:
| [1] |
H. Amann,
Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 229-256.
|
| [2] |
Y. H. Lee and I. Sim,
Global bifurcation phenomena for singular one-dimensional p-Laplacian, J. Differential Equations, 229 (2006), 620-709.
|
| [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.
|
| [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. |
| [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.
|
Figure 2.
Bifurcation for all $\lambda>0.$
Figure 3.
Multiplicity results for certain range of $\lambda$ .
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