Browsing by Subject "fractional operator"

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Access status: Open Access ,
Fractional operators and their commutators on generalized Orlicz spaces
(Wydawnictwa AGH, 2022) Karppinen, Arttu
In this paper we examine boundedness of fractional maximal operator. The main focus is on commutators and maximal commutators on generalized Orlicz spaces (also known as Musielak-Orlicz spaces) for fractional maximal functions and Riesz potentials. We prove their boundedness between generalized Orlicz spaces and give a characterization for functions of bounded mean oscillation.
Access status: Open Access ,
Study of fractional semipositone problems on RN
(Wydawnictwa AGH, 2024) Biswas, Nirjan
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 $r \in [\frac{2N}{N-2s}, \infty]$.