Liouville type theorem for some nonlocal elliptic equations

Yu, Xiaohui

doi:10.1016/j.jde.2017.07.028

Cited by 4 publications

(1 citation statement)

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“…Instead of using the maximum principles, we use some integral inequalities to substitute the maximum principles. This method is usually called the moving plane method in an integral form, we refer the readers to [8,9,10,12,13,14] and etc. During the proof of Theorem 1.2, the following Hardy-Littlewood-Sobolev inequality [5] on half space will be used.…”

Section: (Iii) At Least One Of the Functionsmentioning

confidence: 99%

Liouville type theorem for Hartree-Fock Equation on half space

Chen

2022

CPAA

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In this paper, we study the Liouville type theorem for the following Hartree-Fock equation in half space<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \begin{cases} - \Delta {u_i}(y) = \sum\limits_{j = 1}^n {{\int _{\partial \mathbb{R}_ + ^N}}} \frac{{{u_j}(\bar x, 0){F_1}({u_j}(\bar x, 0))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}d\bar x{f_2}({u_i}(y)) \\ \qquad \qquad \qquad + \sum\limits_{j = 1}^n {{\int _{\partial \mathbb{R}_ + ^N}}} \frac{{{u_j}(\bar x, 0){F_2}({u_i}(\bar x, 0))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}d\bar x{f_1}({u_j}(y)), \ y \in \mathbb{R}_ + ^N, \hfill \\ \frac{{\partial {u_i}}} {{\partial \nu }}(\bar x, 0) = \sum\limits_{j = 1}^n {{\int _{ \mathbb{R}_ + ^N}}} \frac{{{u_j}(y){G_1}({u_j}(y))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}dy{g_2}({u_i}(\bar x, 0)) \\ \qquad \qquad \qquad + \sum\limits_{j = 1}^n {{\int _{ \mathbb{R}_ + ^N}}} \frac{{{u_j}(y){G_2}({u_i}(y))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}dy{g_1}({u_j}(\bar x, 0)), \quad \quad(\bar x, 0) \in \partial \mathbb{R}_ + ^N, \end{cases} \end{align*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}_+^N = \{x\in{\mathbb{R}^N}: x_N > 0\}, f_1, f_2, g_1, g_2, F_1, F_2, G_1, G_2 $\end{document}</tex-math></inline-formula> are some nonlinear functions. Under some assumptions on the nonlinear functions <inline-formula><tex-math id="M2">\begin{document}$ F, G, f, g $\end{document}</tex-math></inline-formula>, we will prove the above equation only possesses trivial positive solution. We use the moving plane method in an integral form to prove our result.

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Section: (Iii) At Least One Of the Functionsmentioning

confidence: 99%

Liouville type theorem for Hartree-Fock Equation on half space

Chen

2022

CPAA

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Liouville type theorems for two mixed boundary value problems with general nonlinearities

2018

Journal of Mathematical Analysis and Applications

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In this paper, we study the nonexistence of positive solutions for the following two mixed boundary value problems. The first problem is the mixed nonlinear-Neumann boundary value problem where R N + = {x ∈ R N : xN > 0}, Γ1 = {x ∈ R N : xN = 0, x1 < 0} and Γ0 = {x ∈ R N : xN = 0, x1 > 0}. We will prove that these problems possess no positive solution under some assumptions on the nonlinear terms. The main technique we use is the moving plane method in an integral form.keywords: Liouville type theorem, Moving plane method, Maximum principle, Mixed boundary value.

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Liouville type theorems for Hartree and Hartree–Fock equations

Jiang

2019

Nonlinear Analysis

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Liouville type theorem for some nonlocal elliptic equations

Abstract: In this paper, we prove some Liouville theorem for the following elliptic equations involving nonlocal nonlinearity and nonlocal boundary value condition

Cited by 4 publications

References 32 publications

Liouville type theorem for Hartree-Fock Equation on half space

Liouville type theorem for Hartree-Fock Equation on half space

Liouville type theorems for two mixed boundary value problems with general nonlinearities

Liouville type theorems for Hartree and Hartree–Fock equations

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