Weighted Liouville Theorems and A Priori Bounds for Indefinite Superlinear Equations with Pucci's Maximal Operator
Year: 2026
Type: preprint
Abstract: Consider \begin{equation*}\ \left\{ \begin{aligned} -\mathcal_{λ,Λ}^{+}(D^{2}u)&=μu+a(x)u^{p}\quad\text{in}~Ω,\\ u&=0\quad\text{on}~~\partialΩ, \end{aligned} \right. \end{equation*} where $ \ Omega$ is a smooth bounded domain, $a$ is a continuous function which is allowed to change sign. Moreover, the set of zeros of $a$ may have a non-empty interior. The main objective of this article is to establish a uniform bound on the set of positive solutions of the above problem. This approach requires a suitable Liouville-type theorem. We first prove the Liouville-type theorem for $-\mathcal_{λ,Λ}^{+}(D^{2}v)=h(x_{1})f(v),$ where \(h\) is a strictly increasing one-dimensional weight and \(f\) is positive and nondecreasing.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221572354