A priori bounds for superlinear elliptic equations with semidefinite nonlinearity

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• 作者：Yūki Naito^a ; ^{ynaito@ehime-u.ac.jp} ; Takashi Suzuki^b ; ^{suzuki@sigmath.es.osaka-u.ac.jp} ; Yohei Toyota^b ; ^{y-toyota@sigmath.es.osaka-u.ac.jp}
• 关键词：primary ; 35B45 ; 35B53 ; 35J60 ; secondary ; 35K55
• 刊名：Nonlinear Analysis: Theory, Methods & Applications
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：151
• 期：Complete
• 页码：18-40
• 全文大小：751 K
• 卷排序：151

文摘

We derive a priori bounds for positive solutions of the superlinear elliptic problems −Δu=a(x)up−Δu=a(x)up on a bounded domain ΩΩ in RN, where a(x)a(x) is Hölder continuous in ΩΩ. Our main motivation is to study the case where a(x)≥0a(x)≥0, a(x)≢0a(x)≢0 and a(x)a(x) has some zero sets in ΩΩ. 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−Δu=A(x′)up in RN, where AA is the continuous function defined on the subspace Rk with 1≤k≤N1≤k≤N and x′∈Rk. We also establish a priori bounds of global nonnegative solutions to the corresponding parabolic initial–boundary value problems.