Computational method to obtain positive solution for classes of Laplacian systems with sign-changing weight functions

摘要

Using a numerical method based on sub-supersolution, we will obtain positive solution to the coupled-system of boundary value problems of the form-Δu(x)=λF(x,u,v),x∈Ω,-Δv(x)=λH(x,u,v),x∈Ω,u(x)=0=v(x),x∈∂Ω,whereF(x,u,v)=[g(x)a(u)+f(v)],H(x,u,v)=[g(x)b(v)+h(u)],λ>0 is a parameter, Δ is the Laplacian operator, Ω is a bounded region in Rn (N ⩾ 1) with smooth boundary ∂Ω and g is a C1 sign-changing function that may be negative near the boundary and f,h,a,b are C1 nondecreasing function satisfying a(0) ⩾ 0, b(0) ⩾ 0lims→∞a(s)s=0,lims→∞b(s)s=0,lims→∞f(s)s=∞,lims→∞h(s)s=∞andlims→∞f(Mh(s))s=0∀M>0.