Liouville theorems for nonnegative solutions to Hardy–Hénon type system on a half space

2021 ◽  
Vol 13 (1) ◽  
Author(s):  
Wei Dai ◽  
Shaolong Peng
Keyword(s):  
Half Space ◽  
Type System ◽  
Liouville Theorems ◽  
Nonnegative Solutions

Solution of one overdetermined stationary Stokes-type system in the half-space

2021 ◽  
Vol 24 (4) ◽  
pp. 54-63
Author(s):  
Kh. Kh. Imomnazarov ◽  
Sh. Kh. Imomnazarov ◽  
M. V. Urev ◽  
R. Kh. Baxromov
Keyword(s):  
Half Space ◽  
Type System

scholarly journals Strong Comparison Principle for the Fractional p-Laplacian and Applications to Starshaped Rings

2018 ◽  
Vol 18 (4) ◽  
pp. 691-704 ◽  
Author(s):  
Sven Jarohs
Keyword(s):  
Half Space ◽  
Comparison Principle ◽  
Nonnegative Function ◽  
Open Set ◽  
Nonnegative Solutions ◽  
Strong Comparison Principle

AbstractIn the following, we show the strong comparison principle for the fractional p-Laplacian, i.e. we analyze\quad\left\{\begin{aligned} \displaystyle(-\Delta)^{s}_{p}v+q(x)\lvert v\rvert% ^{p-2}v&\displaystyle\geq 0&&\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle(-\Delta)^{s}_{p}w+q(x)\lvert w\rvert^{p-2}w&\displaystyle\leq 0&% &\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle v&\displaystyle\geq w&&\displaystyle\phantom{}\text{in ${\mathbb% {R}^{N}}$},\end{aligned}\right.where {s\in(0,1)}, {p>1}, {D\subset\mathbb{R}^{N}} is an open set, and {q\in L^{\infty}(\mathbb{R}^{N})} is a nonnegative function. Under suitable conditions on s, p and some regularity assumptions on v, w, we show that either {v\equiv w} in {\mathbb{R}^{N}} or {v>w} in D. Moreover, we apply this result to analyze the geometry of nonnegative solutions in starshaped rings and in the half space.

scholarly journals Liouville Theorems for Fractional Parabolic Equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wenxiong Chen ◽  
Leyun Wu
Keyword(s):  
Half Space ◽  
Parabolic Equations ◽  
Parabolic Problems ◽  
Maximum Principles ◽  
Qualitative Properties ◽  
Liouville Type ◽  
Liouville Theorems ◽  
Narrow Region ◽  
Liouville Type Theorems ◽  
Whole Space

Abstract In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u → 0 u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space R n - 1 × R \mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems.

scholarly journals Classification of nonnegative solutions to fractional Schrödinger-Hatree-Maxwell type system

2021 ◽  
Vol 6 (12) ◽  
pp. 13665-13688
Author(s):  
Yaqiong Liu ◽  
◽  
Yunting Li ◽  
Qiuping Liao ◽  
Yunhui Yi
Keyword(s):  
Nonnegative Solution ◽  
Type System ◽  
Maximum Principles ◽  
Nonlocal Nonlinearity ◽  
Narrow Region ◽  
Nonnegative Solutions ◽  
Moving Spheres

<abstract><p>In this paper, we are concerned with the fractional Schrödinger-Hatree-Maxwell type system. We derive the forms of the nonnegative solution and classify nonlinearities by appling a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., <italic>narrow region principle</italic> (Theorem 2.3).</p></abstract>

Symmetry of components and Liouville theorems for noncooperative elliptic systems on the half-space

2014 ◽  
Vol 352 (4) ◽  
pp. 321-325 ◽  
Author(s):  
Alexandre Montaru ◽  
Philippe Souplet
Keyword(s):  
Half Space ◽  
Elliptic Systems ◽  
Liouville Theorems

scholarly journals Liouville theorems on the upper half space

2020 ◽  
Vol 40 (9) ◽  
pp. 5373-5381
Author(s):  
Lei Wang ◽  
◽  
Meijun Zhu ◽  
◽  
Keyword(s):  
Half Space ◽  
Liouville Theorems
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