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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 |
$n\times n$ |
$p$ |
$\begin{equation*}\begin{cases}-\left(\varphi_{p_1}(u_1')\right)' = \lambda h_1(t) \left(u_1^{p_1-1-\alpha_1}+f_1(u_2)\right),\quad t\in (0,1),\\-\left(\varphi_{p_2}(u_2')\right)' = \lambda h_2(t) \left(u_2^{p_2-1-\alpha_2}+f_2(u_3)\right),\quad t\in (0,1),\\\quad\quad\quad\vdots\qquad\,\: =\quad\quad\quad\quad\quad\quad \vdots\\-\left(\varphi_{p_n}(u_n')\right)' = \lambda h_n(t) \left(u_n^{p_n-1-\alpha_n}+f_n(u_1)\right),~~\, t\in (0,1),\\\quad\,\,\,\, u_j(0)=0=u_j(1); ~~ j=1,2,\dots,n, \\ \end{cases}\end{equation*}$ |
$\lambda$ |
$p_j>1$ |
$\alpha_j\in(0,p_j-1)$ |
$\varphi_{p_j}(w)=|w|^{p_j-2}w$ |
$h_j \in C((0,1),(0, \infty))\cap L^1((0,1),(0,\infty))$ |
$j=1,2,\dots,n$ |
$f_j:[0,\infty)\rightarrow[0,\infty)$ |
$j=1,2,\dots,n$ |
$f_j(0)=0$ |
$\lambda>0$ |
$\lambda$ |
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.
|



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