Nonexistence of Positive Solutions for Polyharmonic Systems in ℝN

2007 ◽  
Vol 7 (3) ◽  
Author(s):  
Yuxia Guo ◽  
Jiaquan Liu ◽  
Yajing Zhang
Keyword(s):  
Positive Solutions ◽  
Integral Inequalities ◽  
Hardy’S Inequality ◽  
Liouville Type ◽  
Hardy's Inequality ◽  
Liouville Type Theorems ◽  
Moving Plane

AbstractThis work is devoted to the nonexistence of positive solutions for polyharmonic systems(−Δ)Byusing the method of moving plane combined with integral inequalities and Hardy’s inequality, we prove some new Liouville type theorems for the above semilinear polyharmonic systems in ℝ

On Some Integral Inequalities Related to Hardy's

1977 ◽  
Vol 20 (3) ◽  
pp. 307-312 ◽  
Author(s):  
Christopher Olutunde Imoru
Keyword(s):  
Integral Inequality ◽  
Convex Functions ◽  
Jensen’S Inequality ◽  
Integral Inequalities ◽  
Hardy’S Inequality ◽  
Jensen's Inequality ◽  
Hardy's Inequality ◽  
Special Case

AbstractWe obtain mainly by using Jensen's inequality for convex functions an integral inequality, which contains as a special case Shun's generalization of Hardy's inequality.

scholarly journals A Liouville theorem for the higher-order fractional Laplacian

2019 ◽  
Vol 21 (02) ◽  
pp. 1850005 ◽  
Author(s):  
Ran Zhuo ◽  
Yan Li
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
Liouville Theorem ◽  
Integral Form ◽  
Higher Order ◽  
Order Equation ◽  
Liouville Type ◽  
Liouville Type Theorems ◽  
Higher Order Equation ◽  
Fractional Equation

We study Navier problems involving the higher-order fractional Laplacians. We first obtain nonexistence of positive solutions, known as the Liouville-type theorems, in the upper half-space [Formula: see text] by studying an equivalent integral form of the fractional equation. Then we show symmetry for positive solutions on [Formula: see text] through a delicate iteration between lower-order differential/pseudo-differential equations split from the higher-order equation.

On a new class of Hardy type inequalities

1987 ◽  
Vol 105 (1) ◽  
pp. 265-274 ◽  
Author(s):  
B.G. Pachpatte
Keyword(s):  
Integral Inequalities ◽  
Hardy’S Inequality ◽  
Hardy Type Inequalities ◽  
New Class ◽  
Hardy Type ◽  
Hardy's Inequality

SynopsisIn this paper we establish a new class of integral inequalities which originate from the well-known Hardy's inequality. The analysis used in the proofs is quite elementary and is based on the idea used by Levinson to obtain generalisations of Hardy's inequality.

Liouville-type results for positive solutions of pseudo-relativistic Schrödinger system

2021 ◽  
pp. 1-33
Author(s):  
Yuxia Guo ◽  
Shaolong Peng
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Unbounded Domain ◽  
Direct Method ◽  
Strict Monotonicity ◽  
Liouville Type ◽  
Moving Plane ◽  
Liouville Type Results ◽  
Image Position ◽  
Schrödinger System

In this paper, we are concerned with the physically engaging pseudo-relativistic Schrödinger system: \[ \begin{cases} \left(-\Delta+m^{2}\right)^{s}u(x)=f(x,u,v,\nabla u) & \hbox{in } \Omega,\\ \left(-\Delta+m^{2}\right)^{t}v(x)=g(x,u,v,\nabla v) & \hbox{in } \Omega,\\ u>0,v>0 & \hbox{in } \Omega, \\ u=v\equiv 0 & \hbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \] where $s,t\in (0,1)$ and the mass $m>0.$ By using the direct method of moving plane, we prove the strict monotonicity, symmetry and uniqueness for positive solutions to the above system in a bounded domain, unbounded domain, $\mathbb {R}^{N}$ , $\mathbb {R}^{N}_{+}$ and a coercive epigraph domain $\Omega$ in $\mathbb {R}^{N}$ , respectively.

scholarly journals On some extensions of Hardy’s inequality

1985 ◽  
Vol 8 (1) ◽  
pp. 165-171 ◽  
Author(s):  
Christopher O. Imoru
Keyword(s):  
Integral Inequalities ◽  
Hardy’S Inequality ◽  
Sharp Focus ◽  
Hardy's Inequality

We present in this paper some new integral inequalities which are related to Hardy's inequality, thus bringing into sharp focus some of the earlier results of the author.

Integral inequalities related to Hardy's inequality

1985 ◽  
Vol 28 (1) ◽  
pp. 199-207 ◽  
Author(s):  
R. N. Mohapatra ◽  
D. C. Russell
Keyword(s):  
Integral Inequalities ◽  
Hardy’S Inequality ◽  
Hardy's Inequality

More on reverses of Minkowski’s inequalities and Hardy’s integral inequalities

2018 ◽  
Vol 13 (03) ◽  
pp. 2050064
Author(s):  
Bouharket Benaissa
Keyword(s):  
Hardy Inequality ◽  
Integral Inequality ◽  
Minkowski Inequality ◽  
Integral Inequalities ◽  
Hardy’S Inequality ◽  
Minkowski’S Inequality ◽  
Hardy's Inequality ◽  
Hardy’S Inequalities

In 2012, Sulaiman [Reverses of Minkowski’s, Hölder’s, and Hardy’s integral inequalities, Int. J. Mod. Math. Sci. 1(1) (2012) 14–24] proved integral inequalities concerning reverses of Minkowski’s and Hardy’s inequalities. In 2013, Banyat Sroysang obtained a generalization of the reverse Minkowski’s inequality [More on reverses of Minkowski’s integral inequality, Math. Aeterna 3(7) (2013) 597–600] and the reverse Hardy’s integral inequality [A generalization of some integral inequalities similar to Hardy’s inequality, Math. Aeterna 3(7) (2013) 593–596]. In this article, two results are given. First one is further improvement of the reverse Minkowski inequality and second is further generalization of the integral Hardy inequality.

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