| For a sign-changing function a(x) ∈ CalocR n with bounded O+ = {x ∈ Rn | a(x) > 0}, we study non-negative entire solutions u(x) ≥ 0 of the semilinear elliptic equation -Deltau = a( x)uq + b( x)up in Rn with n ≥ 3, 0 < q < 1, p > q, and lambda > 0. We consider two types of coefficient b(x) ∈ CalocR n , either b(x) ≤ 0 in Rn , or b(x) ≡ 1. In each case, we give sufficient conditions on a(x) for which all solutions must have compact support. In case O+ 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 a gamma = gammaa+(x) - a-(x). We show that there exists a maximal interval gamma ∈ (0, Gamma] 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 gamma ∈ (0, Gamma). 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. |
