Compactly supported solutions to stationary degenerate diffusion equations

We consider non-negative solutions of the semilinear elliptic equation in RnRn with n⩾3n⩾3:−Δu=a(x)uq+b(x)up,−Δu=a(x)uq+b(x)up, where 0qp>q, a(x)a(x) sign-changing, a=a+−a−a=a+−a− and b(x)⩽0b(x)⩽0 is non-positive. Under appropriate growth assumption on a−a− at infinity, we prove that all solutions in D1,2(Rn)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}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+Ω0+. For instance, we introduce a positive parameter λ and replace a   by λa+−a−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+Ω0+. For λ   large and p⩽1p⩽1 the solution is unique and supported in all of Ω0+Ω0+. We also prove the existence of the limit λ→∞λ→∞ of this solution, which solves −Δw=a+wq−Δw=a+wq and lim|x|→∞w(x)=0lim|x|→∞w(x)=0. The analysis is based on comparison arguments and a priori bounds.