A priori bounds for superlinear elliptic equations with semidefinite nonlinearity. (March 2017) - Naito, Yūki; Suzuki, Takashi; Toyota, Yohei
Abstract: We derive a priori bounds for positive solutions of the superlinear elliptic problems − Δ u = a ( x ) u p on a bounded domain Ω in R N, 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 − Δ u = A ( x ′ ) u p in R N, where A is the continuous function defined on the subspace R k with 1 ≤ k ≤ N and x ′ ∈ R k . We also establish a priori bounds of global nonnegative solutions to the corresponding parabolic initial–boundary value problems.