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Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system

  • * Corresponding author: R. Shivaji.

    * Corresponding author: R. Shivaji. 
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  • 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.

    Mathematics Subject Classification: Primary: 34B16, 34B18; Secondary: 35J57.

    Citation:

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  • Figure 1.  Bifurcation of solution from the origin.

    Figure 2.  Bifurcation for all $\lambda>0.$

    Figure 3.  Multiplicity results for certain range of $\lambda$ .

  •   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. 
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