Finite difference approximate solutions for a mixed sub-superlinear equation

In Ben Mabrouk and Ben Mohamed (2006) [A. Ben Mabrouk, M.L. Ben Mohamed, On some critical and slightly super-critical sub-superlinear equations. Far East J. Appl. Math. 23(1) (2006) 73-90, Special Volume of PDEs; A. Ben Mabrouk, M.L. Ben Mohamed, Nodal solutions for some nonlinear elliptic equations, Appl. Math. Comp., in press, doi:10.1016/j.amc.2006.08.003], it has proved some theoretical results dealing with some boundary value problem Δu + f(u) = 0 in B and u = 0 on ∂B, where B is a domain in Rd with smooth boundary. In the present paper, we perform a difference scheme method to approximate the solution of a nonlinear evolutionary problem associated to the elliptic problem studied in Ben Mabrouk and Ben Mohamed (2006). We give at the end some numerical implementations based on a mixed sublinear superlinear term of the form f(u) = ∣u∣p−1u + λ∣u∣q−1u.