COMPETITION PHENOMENA FOR DIFFERENCE EQUATIONS WITH OSCILLATORY NONLINEARITIES

In this paper, we study the following discrete boundary value problem { −∆(∆u(k − 1)) = λa(k)u(k) + f(u(k)) for all k ∈ [1, T ], u(0) = u(T + 1) = 0, where f : [0,+∞) → R is a continuous function oscillating near the origin or at infinity. By using direct variational methods, we prove that, when f oscillates near the origin, the problem admits a sequence of non-negative, distinct solutions which converges to 0 if p > 1 and at least a finite number of solutions if 0 < p < 1. While, when f oscillates at infinity, the converse holds true, that is, there is a sequence of non-negative, distinct solutions which converges to +∞ if 0 < p 6 1 and at least a finite number of solutions if p > 1. Dedicated with esteem to Professor Enzo L. Mitidieri on his 60th anniversary