Compactly supported solutions to stationary degenerate diffusion equations

We consider non-negative solutions of the semilinear elliptic equation in R-n with n >= 3: -Delta u = a(x)u(q) + b(x)u(P). where 0 < q < 1, p > q, 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 D-1.2(R-n) are compactly supported and their support is contained in a large ball with radius determined by a. When Omega(0+) = {x is an element of R-n | 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 Omega(0+). For instance, we introduce a positive parameter lambda and replace a by lambda a(+) - a(-). We then show that for lambda small, all solutions have compact support and there exist solutions with supports in any combination of these connected components of Omega(0+). For lambda large and p <= 1 the solution is unique and supported in all of Omega(0+). We also prove the existence of the limit lambda -> infinity of this solution, which solves -Delta w = a(+) w(q) and lim(|x|->infinity) w(x) = 0. The analysis is based on comparison arguments and a prior! bounds. (C) 2009 Elsevier Inc. All rights reserved.