Elliptic Systems With Some Superlinear Assumption Only Around the Origin

In this paper, using a priori bound techniques we study existence of positive solutions of the elliptic system: {-div(|x|α1∇u)=|x|β1f(|x|,u,v)x∈B,-div(|x|α2∇v)=|x|β2g(|x|,u,v)x∈B,u(x)=0=v(x),x∈∂B.where B is the unitary ball centered at the origin. Assuming that f, g are nonnegative nonlinearities and that f(| x| , u, v) + g(| x| , u, v) is superlinear at 0 and at ∞, we establish some results of existence of one positive solution. As an application, we establish two positive solutions for some non-homogeneous elliptic system. The main novelties here are that the nonlinearities could have growth above the critical hyperbola on some part of the domain as well as only local superlinear hypotheses at ∞.