@Article{Biswas2024,
author = "Biswas, Nirjan",
title = "Study of fractional semipositone problems on R<sup>N</sup>",
journal = "Opuscula Math.",
year = "2024",
volume = "44",
number = "4",
pages = "445--470",
doi = "https://doi.org/10.7494/OpMath.2024.44.4.445",
url = "https://doi.org/10.7494/OpMath.2024.44.4.445",
abstract = "Let \(s\in (0,1)\) and \(N\gt 2s\). In this paper, we consider the following class of nonlocal semipositone problems: \[(-\Delta)^s u= g(x)f_a(u)\text{ in }\mathbb{R}^N,\quad u \gt 0\text{ in }\mathbb{R}^N,\] where the weight \(g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)\) is positive, \(a\gt 0\) is a parameter, and \(f_a \in \mathcal{C}(\mathbb{R})\) is strictly negative on \((-\infty,0]\). For \(f_a\) having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution \(u_a\), provided \(a\) is near zero. To obtain the positivity of \(u_a\), we establish a Brezis-Kato type uniform estimate of \((u_a)\) in \(L^r(\mathbb{R}^N)\) for every \(r \in [\frac{2N}{N-2s}, \infty]\).",
keywords = "semipositone problems, fractional operator, uniform regularity estimates, positive solutions",
issn = "1232-9274"
}