We discuss the discrete
p-Laplacian eigenvalue problem,
$$left{ begin{gathered} Delta (phi _p (Delta u(k - 1))) + lambda a(k)g(u(k)) = 0,k in { 1,2,...,T} , hfill u(0) = u(T + 1) = 0, hfill end{gathered} right.$$
where
T > 1 is a given positive integer and
φ p (
x):= |
x|
p?2 x,
p > 1. First, the existence of an unbounded continuum
C of positive solutions emanating from (
λ,
u) = (0, 0) is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any
λ > 0 and all solutions are ordered. Thus the continuum
C is a monotone continuous curve globally defined for all
λ > 0.
