Liouville Type Results For Polyharmonic Inequalities with Nonlocal Terms

Bei Wang *

School of Mathematics and Statistics, Southwest University, Chongqing 400715, People's Republic of China.

*Author to whom correspondence should be addressed.


Abstract

In this note, we study the polyharmonic inequalities system
\[
(-\Delta)^m u_i \geq \sum_{j=1}^n e_{i j}\left(\Psi_{i j}(|x|) * u_j^{p_{i j}}\right) u_i^{q_{i j}} \quad \text { in }{ }^N, \quad i=1,2, \cdots, n,
\]
where \(N \geq 1\) and \(m \geq 1\) are integers, \(p_{i j} \geq 1, q_{i j}>0\). \(\Delta^m\) denotes the m-polyharmonic operator. The operator \(*\) denotes the convolution and \(\Psi_{i j}\) is a function that has certain properties. \(\left(e_{i j}\right)\) is the adjacency matrix. By poly-superharmonic propery of u and some estimates, we get a Liouville type result of (0.1), which generalize the recent results on these inequalities.

Keywords: Polyharmonic inequalities, nonlocal terms, liouville type results


How to Cite

Wang, B. (2023). Liouville Type Results For Polyharmonic Inequalities with Nonlocal Terms. Asian Journal of Mathematics and Computer Research, 30(2), 9–16. https://doi.org/10.56557/ajomcor/2023/v30i28236

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References

Marius Ghergu, Yasuhito Miyamoto, Vitaly Moroz. Polyharmonic inequalities with nonlocal terms. J. Differential Equations. 2021;296:799-821.

Peng, Shaolong. Existence and Liouville theorems for coupled fractional elliptic system with Stein-Weiss type convolution parts. Mathematische Zeitschrift. 2022;302(3):1593-1626.

Dai, Wei, Peng, Shaolong, Qin, Guolin. Liouville type theorems, a priori estimates and existence of solutions for sub-critical order Lane Emden Hardy equations. Journal d'Analyse Mathematique. 2022;146(2):673-718.

Cowan, Craig, Razani, Abdolrahman. Singular solutions of a Henon equation involving a nonlinear gradient term. Communications on Pure & Applied Analysis. 2022;21(1).

I. Pekars. Untersuchungen uber die Elektronentheorie der Kristalle. Untersuchungen uber die Elektronentheorie der Kristalle; 1954.

Krw Jones. Newtonian quantum gravity. Australian Journal of Physics; 1995.

Irene M. Moroz, Roger Penrose, Paul Tod. Spherically-symmetric solutions of the Schrodinger-Newton equations. Topology of the Universe Conference (Cleveland, OH, 1997). 1998;15:2733-2742.

Vitaly Moroz, Jean Van Schaftingen. A guide to the Choquard equation. J. Fixed Point Theory Appl. 2017;19(1):773-813.

Marius Ghergu, Paschalis Karageorgis, Gurpreet Singh. Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms. J. Di erential Equations. 2020;268(10):6033-6066.

Vitaly Moroz, Jean Van Schaftingen. Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains. J. Di erential Equations. 2013;254(8):3089-3145.

Bidaut-Veron, Marie Francoise and Giacomini, Hector. A new dynamical approach of Emden-Fowler equations and systems. ADV.Di erential Equation. 2010;15:1033-1082.

Cao, Daomin, Dai, Wei. Classi cation of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 2019;149(4):979-994.

Chen, Wenxiong, Fang, Yanqin, Yang, Ray. Liouville theorems involving the fractional Laplacian on a half space. Advances in Mathematics. 2015;274:167-198.

Cheng, Tingzhi, Liu, Shuang. A Liouville type theorem for higher order Hardy{Henon equation in RN. Journal of Mathematical Analysis and Applications. 2016;444(1):370-389.

Dai, Wei, Qin, Guolin. Liouville type theorems for Hardy{Henon equations with concave nonlinearities. Mathematische Nachrichten. 2020;293(6):1084-1093.

Dai, Wei, Qin, Guolin. Classi cation of nonnegative classical solutions to third-order equations. Advances in Mathematics. 2018;328:822-857.