Compactly supported solutions to stationary degenerate diffusion equations

We consider non-negative solutions of the semilinear elliptic equation in Rn with n⩾3:−Δu=a(x)uq+b(x)up, where 0q, a(x) sign-changing, a=a+−a− and b(x)⩽0 is non-positive. Under appropriate growth assumption on a− at infinity, we prove that all solutions in D1,2(Rn) are compactly supported and their support is contained in a large ball with radius determined by a. When Ω0+={x∈Rn|a(x)⩾0} has several compact connected components, we give …