# Nonlinear Inequalities with Double Riesz Potentials

@article{Ghergu2021NonlinearIW,
title={Nonlinear Inequalities with Double Riesz Potentials},
author={Marius Ghergu and Zeng Liu and Yasuhito Miyamoto and Vitaly Moroz},
journal={Potential Analysis},
year={2021}
}
• Published 7 June 2021
• Physics
• Potential Analysis
<jats:p>We investigate the nonnegative solutions to the nonlinear integral inequality <jats:italic>u</jats:italic> ≥ <jats:italic>I</jats:italic><jats:sub><jats:italic>α</jats:italic></jats:sub> ∗((<jats:italic>I</jats:italic><jats:sub><jats:italic>β</jats:italic></jats:sub> ∗ <jats:italic>u</jats:italic><jats:sup><jats:italic>p</jats:italic></jats:sup>)<jats:italic>u</jats:italic><jats:sup><jats:italic>q</jats:italic></jats:sup>) a.e. in <jats:inline-formula><jats:alternatives><jats:tex-math…

## References

SHOWING 1-10 OF 21 REFERENCES
Classification of solutions for an integral equation
• Mathematics
• 2006
Let n be a positive integer and let 0 < α < n. Consider the integral equation $$(0.1) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; u(x) = \int\limits^{}_{R^{n}} {1 \over |x -y|^{n-\alpha}}u(y)^{{n+\alpha} \over Representation Formulae for Solutions to Some Classes of Higher Order Systems and Related Liouville Theorems • Mathematics • 2008 Abstract.Let m ≥ 1 be an integer and N > 2m. Let μ be a positive Radon measure on$${\mathbf{R}}^N$$. We study necessary and sufficient conditions on possible distributional solutions of Liouville theorems for an integral equation of Choquard type We establish sharp Liouville theorems for the integral equation \begin{document} u(x) = \int_{\mathbb{R}^n} \frac{u^{p-1}(y)}{|x-y|^{n-\alpha}} \int_{\mathbb{R}^n} \frac{u^p(z)}{|y-z|^{n-\beta}} dz Qualitative properties of solutions for an integral equation • Mathematics • 2003 Let n be a positive integer and let  0 < \alpha < n. In this paper, we study more general integral equation  u(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} K(y) u(y)^p dy. We establish Asymptotic behavior at isolated singularities for solutions of nonlocal semilinear elliptic systems of inequalities • Mathematics • 2014 We study the behavior near the origin of$$C^2$$C2 positive solutions$$u(x)$$u(x) and$$v(x)$$v(x) of the system$$\begin{aligned} \left\{ \begin{aligned} 0\le -\Delta u\le \left(
Nonlocal Hardy type inequalities with optimal constants and remainder terms
• Mathematics
• 2012
Using a groundstate transformation, we give a new proof of the optimal Stein--Weiss inequality of Herbst [ int_{R^N} int_{R^N} rac{arphi (x)}{abs{x}^rac{alpha}{2}} I_alpha (x - y) rac{arphi
Positive powers of the Laplacian: from hypersingular integrals to boundary value problems
• Mathematics
• 2017
Any positive power of the Laplacian is related via its Fourier symbol to a hypersingular integral with finite differences. We show how this yields a pointwise evaluation which is more flexible than