GLOBAL CONTINUUM OF POSITIVE SOLUTIONS FOR DISCRETE p-LAPLACIAN EIGENVALUE PROBLEMS

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