Nonconstant radial positive solutions of elliptic systems with Neumann boundary conditions

Let BRBR be the ball of radius R   in RNRN with N≥2N≥2. We consider the nonconstant radial positive solutions of elliptic systems of the form−Δu+u=f(u,v)inBR,−Δv+v=g(u,v)inBR,∂νu=∂νv=0on∂BR, where f and g are nondecreasing in each component. With few assumptions on the nonlinearities, we apply bifurcation theory to show the existence of at least one nonnegative, nonconstant and nondecreasing solution.