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: |
H. Amann
, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976)
, 229-256.
![]() |
|
Y. H. Lee
and I. Sim
, Global bifurcation phenomena for singular one-dimensional p-Laplacian, J. Differential Equations, 229 (2006)
, 620-709.
![]() |
|
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.
![]() |
|
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.
![]() |
|
R. Shivaji
and B. Son
, Bifurcation and multiplicity results for classes of $p,q$ Laplacian systems, Topol. Methods Nonlinear Anal., 48 (2016)
, 103-114.
![]() |
Bifurcation of solution from the origin.
Bifurcation for all
Multiplicity results for certain range of