A priori bounds for superlinear elliptic equations with semidefinite nonlinearity

Abstract

We derive a priori bounds for positive solutions of the superlinear elliptic problems −Δu=a(x)up on a bounded domain Ω in RN, where a(x) is Hölder continuous in Ω. Our main motivation is to study the case where a(x)≥0, a(x)≢0 and a(x) has some zero sets in Ω. We show that, in this case, the scaling arguments reduce the problem of a priori bounds to the Liouville-type results for the equation −Δu=A(x′)up in RN, where A is the continuous function defined on the subspace Rk with 1≤k≤N and x′∈Rk. We also establish a priori bounds of global nonnegative solutions to the corresponding parabolic initial–boundary value problems.