Compact Support and Dead Cores for Stationary Degenerate Diffusion Equations

Abstract

For a sign-changing function a(x) E C^αloc(Rn) with bounded Ω+ = {x E R^n |a(x) > O}, we study non-negative entire solutions u(x) ≥ 0 of the semilinear elliptic equation -Δu = a(x)u^q + b(x)u^p in R^n with n ≥ 3.0 < q < 1, p > q, and λ > 0. We consider two types of coefficient b(x) E C^αloc(R^n), either b(x) ≤ 0 in (R^n) or b(x) ≡ 1. In each case, we give sufficient conditions on a(x) for which all solutions must have compact support. In case Ω+ has several connected components, we also give conditions under which there exist "dead core'' solutions which vanish identically in one or more of these components. In the "logistic" case b(x) ≤ 0, we prove that there can be only one solution with given dead core components. In the case b(x) ≡ 1, the question of existence is more delicate, and we introduce a parametrized family of equations by replacing a(x) by ay = ya^+(x) - a^- (x). We show that there exists a maximal interval y E (0, f] for which there exists a stable (locally minimizing) solution. Under some hypotheses on a^- near infinity, we prove that there are two solutions for each y E (0, f). Some care must be taken to ensure the compactness of Palais-Smale sequences, and we present an example which illustrates how the Palais-Smale condition could fail for certain a(x). The analysis is based on a combination of comparison arguments, a priori estimates, and variational methods.