Multiple positive solutions of singular discrete \(p\)-Laplacian problems via variational methods. (English) Zbl 1098.39001

Agarwal, Ravi P.; Perera, Kanishka; O’Regan, Donal

Adv. Difference Equ. 2005, No. 2, 93-99 (2005).

The authors consider the following boundary value problem \[ -\Delta\biggl( \varphi_p\bigl(\Delta u(k-1)\bigr)\biggr)=f\bigl(k,u(k)\bigr), \quad k\in[1,n] \]
\[ u(k)>0, \quad k\in [1,n],\quad u(0)=0=u(n+1) \] where \(n\) is an integer greater or equal than \(1,[1,n]\) is the discrete interval \(\{1,2,\dots,n\}\), \(\Delta u(k)= u(k+1)-u(k)\) is the forward difference operator, \(\varphi_p(s)=|s|^{p-2}s,1< p<\infty\) and \(f\in C([1,n] \times(0,\infty))\) satisfies \[ a_0(k)\leq f(k,t)\leq a_1(k)t^{-\gamma}, \quad (k,t)\in[1,n]\times(0,t_0) \] for some nontrivial functions \(a_0\), \(a_1\geq 0\) and \(\gamma\), \(t_0>0\), so that it may be singular at \(t=0\) and may change sign. Multiple positive solutions for the above problem are found using variational methods.

Reviewer: Stefan Balint (Timişoara)

MSC:

39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis

Keywords:

boundary value problem; multiple positive solutions; variational methods