Báo Cáo Báo cáo hóa học MULTIPLE POSITIVE SOLUTIONS OF SINGULAR DISCRETE p-LAPLACIAN PROBLEMS VIA VARIATIONA

MULTIPLE POSITIVE SOLUTIONS OF SINGULAR DISCRETE p-LAPLACIAN PROBLEMS VIA VARIATIONAL METHODS RAVI P. AGARWAL, KANISHKA PERERA, AND DONAL O’REGAN Received 31 March 2005

We obtain multiple positive solutions of singular discrete p-Laplacian problems using variational methods. 1. Introduction We consider the boundary value problem ư∆ ϕ p ∆u(k ư 1) = f k,u(k) , k ∈ [1,n], (1.1) u(k) 0, k ∈ [1,n], u(0) = 0 = u(n + 1), where n is an integer greater than or equal to 1, [1,n] is the discrete interval {1, .,n}, ∆u(k) = u(k + 1) ư u(k) is the forward difference operator, ϕ p (s) = |s| pư2 s, 1 0, so that it may be singular at t = 0 and may change sign. Let λ1 ,ϕ1 0 be the first eigenvalue and eigenfunction of ư∆ ϕ p ∆u(k ư 1) = λϕ p u(k) , k ∈ [1,n],