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 conditions under which there may or may not exist solutions which vanish identically on one or more of the components of Ω0+. For instance, we introduce a positive parameter λ and replace a by λa+−a−. We then show that for λ small, all solutions have compact support and there exist solutions with supports in any combination of these connected components of Ω0+. For λ large and p⩽1 the solution is unique and supported in all of Ω0+. We also prove the existence of the limit λ→∞ of this solution, which solves −Δw=a+wq and lim|x|→∞w(x)=0. The analysis is based on comparison arguments and a priori bounds.