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Weighted Liouville Theorems and A Priori Bounds for Indefinite Superlinear Equations with Pucci's Maximal Operator
Abstract
Consider { −_λ,Λ+(D2u)&=μu+a(x)up in Ω, u&=0 on ∂Ω, . 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 −_λ,Λ+(D2v)=h(x1)f(v), where h is a strictly increasing one-dimensional weight and f is positive and nondecreasing.
Related
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BibTeX
@misc{rathore2026weighted,
title = {{Weighted Liouville Theorems and A Priori Bounds for Indefinite Superlinear Equations with Pucci's Maximal Operator}},
author = {Karan Rathore and Mohan Mallick and Ram Baran Verma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13486},
eprint = {2610.13486},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13486},
}
RIS
TY - UNPB
TI - Weighted Liouville Theorems and A Priori Bounds for Indefinite Superlinear Equations with Pucci's Maximal Operator
AU - Karan Rathore
AU - Mohan Mallick
AU - Ram Baran Verma
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13486
UR - https://arxiv.org/abs/2610.13486
AB - Consider { −_λ,Λ^+(D^2u)&=μu+a(x)u^p in Ω, u&=0 on ∂Ω, . 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 −_λ,Λ^+(D^2v)=h(x_1)f(v), where h is a strictly increasing one-dimensional weight and f is positive and nondecreasing.
ER -