Existence results for classes of Laplacian systems with sign-changing weight Original Research Article

Consider the system View the MathML source{−Δu=λF(x,u,v),in Ω,−Δv=λH(x,u,v),in Ω,u=0=v,on ∂Ω, Turn MathJax on where F(x,u,v)=[g(x)a(u)+f(v)],H(x,u,v)=[g(x)b(v)+h(u)],λ>0F(x,u,v)=[g(x)a(u)+f(v)],H(x,u,v)=[g(x)b(v)+h(u)],λ>0 is a parameter, ΩΩ is a bounded domain in RN;N≥1RN;N≥1, with smooth boundary ∂Ω∂Ω and View the MathML sourceΔ is the Laplacian operator. Here gg is a C1C1 sign-changing function that may be negative near the boundary and f,h,a,bf,h,a,b are C1C1 nondecreasing functions satisfying a(0)≥0,b(0)≥0a(0)≥0,b(0)≥0, View the MathML sourcelims→∞a(s)s=0,lims→∞b(s)s=0, Turn MathJax on View the MathML sourcelims→∞f(s)=∞,lims→∞h(s)=∞and Turn MathJax on View the MathML sourcelims→∞f(Mh(s))s=0,∀M>0. Turn MathJax on We discuss the existence of positive solutions when f,h,a,bf,h,a,b and gg satisfy certain additional conditions. We employ the method of sub–super-solutions to obtain our results. Note that we do not require any sign-changing conditions on f(0)f(0) or h(0)h(0). We also note that while aa and bb are assumed to be sublinear at ∞∞, we only assume a combined sublinear effect of ff and hh at ∞∞.