Radial positive solutions of elliptic systems with Neumann boundary conditions

We consider radial solutions of elliptic systems of the form{−Δu+u=a(|x|)f(u,v)in BR,−Δv+v=b(|x|)g(u,v)in BR,∂νu=∂νv=0on ∂BR, where essentially a, b are assumed to be radially nondecreasing weights and f, g are nondecreasing in each component. With few assumptions on the nonlinearities, we prove the existence of at least one couple of nondecreasing nontrivial radial solutions. We emphasize that we do not assume any variational structure nor subcritical growth on the nonlinearities. Our result covers systems with supercritical as well as asymptotically linear nonlinearities.