Existence theory for single and multiple solutions to singular positone discrete Dirichlet boundary value problems to the one-dimension $p$-Laplacian

Daqing Jiang, Lili Zhang, Donal O'Regan and Ravi P. Agarwal

Address.
  Department of Mathematics, Northeast Normal University, Changchun 130024, P. R. China

Department of Mathematics, National University of Ireland, Galway, Ireland

Department of Mathematical Science, Florida Institute of Technology, Melbourne, Florida 32901-6975, USA
 

E-mail. agarwal@fit.edu

Abstract.
In this paper we establish the existence of single and multiple solutions to the positone discrete Dirichlet boundary value problem $$ \left\{ \begin{array}{l} \Delta\big[\phi (\Delta u(t-1))\big]+ q(t) f(t,u(t))=0\,,\quad t\in \{1,2,\dots,T\}\\[3pt] u(0)=u(T+1)=0\,, \end{array} \right. $$ where $\phi(s) = |s|^{p-2}s$, $p>1$ and our nonlinear term $f(t,u)$ may be singular at $u=0$.

AMSclassification. 34B15.

Keywords.  Multiple solutions, singular, existence, discrete boundary value problem.