Properties of the first eigenvalue with sign-changing weight of the discrete p-Laplacian and applications

Résumé

By establishing some results around the first eigenvalue λ1(m) for the following problem: -Δ(φp(Δu(k - 1)))= λm(k)φp(u(k)); k∈ [1; n]; u(0) = 0 = u(n + 1); where m ∈ M([1; n]) = {m : [1; n] → R /∃ k∈ [1; n]; m(k) > 0} ; as the constant sign of the first eigenfunction with λ1(m); the simplicity of λ1(m); the strict monotonicity property with respect the weight and sign change of any eigenfunction with  ( λ > λ1(m)); we prove the existence and non-existence of solutions of the problem (1.1).