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

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.

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Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians

Transactions of the American Mathematical Society ◽

10.1090/tran/8389 ◽

2021 ◽

pp. 1

Author(s):

Daomin Cao ◽

Wei Dai ◽

Guolin Qin

Keyword(s):

Higher Order ◽

Solutions To Equations ◽

Liouville Theorems ◽

Nonnegative Solutions

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Liouville Theorems for Fractional Parabolic Equations

Advanced Nonlinear Studies ◽

10.1515/ans-2021-2148 ◽

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.

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Liouville theorems for nonnegative solutions to static weighted Schrödinger–Hartree–Maxwell type equations with combined nonlinearities

Analysis and Mathematical Physics ◽

10.1007/s13324-021-00479-3 ◽

2021 ◽

Vol 11 (2) ◽

Author(s):

Wei Dai ◽

Shaolong Peng

Keyword(s):

Liouville Theorems ◽

Nonnegative Solutions

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

AIMS Mathematics ◽

10.3934/math.2021794 ◽

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>

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