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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

* Corresponding author: R. Shivaji.

Received  March 2016 Revised  August 2016 Published  January 2018

In this paper we study the positive solutions to the
$n\times n$
$p$
-Laplacian system:
$\begin{equation*}\begin{cases}-\left(\varphi_{p_1}(u_1')\right)' = \lambda h_1(t) \left(u_1^{p_1-1-\alpha_1}+f_1(u_2)\right),\quad t\in (0,1),\\-\left(\varphi_{p_2}(u_2')\right)' = \lambda h_2(t) \left(u_2^{p_2-1-\alpha_2}+f_2(u_3)\right),\quad t\in (0,1),\\\quad\quad\quad\vdots\qquad\,\: =\quad\quad\quad\quad\quad\quad \vdots\\-\left(\varphi_{p_n}(u_n')\right)' = \lambda h_n(t) \left(u_n^{p_n-1-\alpha_n}+f_n(u_1)\right),~~\, t\in (0,1),\\\quad\,\,\,\, u_j(0)=0=u_j(1); ~~ j=1,2,\dots,n, \\ \end{cases}\end{equation*}$
where
$\lambda$
is a positive parameter,
$p_j>1$
,
$\alpha_j\in(0,p_j-1)$
,
$\varphi_{p_j}(w)=|w|^{p_j-2}w$
, and
$h_j \in C((0,1),(0, \infty))\cap L^1((0,1),(0,\infty))$
for
$j=1,2,\dots,n$
. Here
$f_j:[0,\infty)\rightarrow[0,\infty)$
,
$j=1,2,\dots,n$
are nontrivial nondecreasing continuous functions with
$f_j(0)=0$
and satisfy a combined sublinear condition at infinity. We discuss here a bifurcation result, an existence result for
$\lambda>0$
, and a multiplicity result for a certain range of
$\lambda$
. We establish our results through the method of sub-super solutions.
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.

[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 1.  Bifurcation of solution from the origin.
Figure 2.  Bifurcation for all $\lambda>0.$
Figure 3.  Multiplicity results for certain range of $\lambda$ .
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