CPAA
Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$
Yanqin Fang Jihui Zhang
Communications on Pure & Applied Analysis 2013, 12(2): 663-678 doi: 10.3934/cpaa.2013.12.663
Let $n, m$ be a positive integer and let $R_+^n$ be the $n$-dimensional upper half Euclidean space. In this paper, we study the following integral equation \begin{eqnarray} u(x)=\int_{R_+^n}G(x,y)u^pdy, \end{eqnarray} where \begin{eqnarray*} G(x,y)=\frac{C_n}{|x-y|^{n-2m}}\int_0^{\frac{4x_n y_n}{|x-y|^2}}\frac{z^{m-1}}{(z+1)^{n/2}}dz, \end{eqnarray*} $C_{n}$ is a positive constant, $0 <2m 1$. Using the method of moving planes in integral forms, we show that equation (1) has no positive solution.
keywords: monotonicity. moving planes in integral forms Nonexistence
CPAA
Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system
Yanqin Fang Jihui Zhang
Communications on Pure & Applied Analysis 2011, 10(4): 1267-1279 doi: 10.3934/cpaa.2011.10.1267
In this paper, we study the following system

$-\epsilon^2\Delta v+V(x)v+\psi(x)v=v^p, \quad x\in R^3,$

$-\Delta\psi=\frac{1}{\epsilon}v^2,\quad \lim_{|x|\rightarrow\infty}\psi(x)=0,\quad x\in R^3,$

where $\epsilon>0$, $p\in (3,5)$, $V$ is positive potential. We relate the number of solutions with topology of the set where $V$ attain their minimum value. By applying Ljusternik-Schnirelmann theory, we prove the multiplicity of solutions.

keywords: multiple solutions. Ljusternik-Schnirelmann theory Schrödinger-Maxwell
DCDS
Regularity and classification of solutions to static Hartree equations involving fractional Laplacians
Wei Dai Jiahui Huang Yu Qin Bo Wang Yanqin Fang
Discrete & Continuous Dynamical Systems - A 2018, 0(0): 1-15 doi: 10.3934/dcds.2018117

In this paper, we are concerned with the fractional order equations (1) with Hartree type $ \dot{H}^{\frac{α}{2}} $-critical nonlinearity and its equivalent integral equations (3). We first prove a regularity result which indicates that weak solutions are smooth (Theorem 1.2). Then, by applying the method of moving planes in integral forms, we prove that positive solutions $ u $ to (1) and (3) are radially symmetric about some point $ x_{0}∈\mathbb{R}^{d} $ and derive the explicit forms for $ u $ (Theorem 1.3 and Corollary 1). As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities (Corollary 2).

keywords: Fractional Laplacians positive solutions radial symmetry uniqueness regularity Hartree type nonlinearity methods of moving planes in integral forms
CPAA
Regularity and nonexistence of solutions for a system involving the fractional Laplacian
De Tang Yanqin Fang
Communications on Pure & Applied Analysis 2015, 14(6): 2431-2451 doi: 10.3934/cpaa.2015.14.2431
We consider a system involving the fractional Laplacian \begin{eqnarray} \left\{ \begin{array}{ll} (-\Delta)^{\alpha_{1}/2}u=u^{p_{1}}v^{q_{1}} & \mbox{in}\ \mathbb{R}^N_+,\\ (-\Delta)^{\alpha_{2}/2}v=u^{p_{2}}v^{q_{2}} &\mbox{in}\ \mathbb{R}^N_+,\\ u=v=0,&\mbox{in}\ \mathbb{R}^N\backslash\mathbb{R}^N_+, \end{array} \right. \end{eqnarray} where $\alpha_{i}\in (0,2)$, $p_{i},q_{i}>0$, $i=1,2$. Based on the uniqueness of $\alpha$-harmonic function [9] on half space, the equivalence between (1) and integral equations \begin{eqnarray} \left\{ \begin{array}{ll} u(x)=C_{1}x_{N}^{\frac{\alpha_{1}}{2}}+\displaystyle\int_{\mathbb{R}_{+}^{N}}G^{1}_{\infty}(x,y)u^{p_{1}}(y)v^{q_{1}}(y)dy,\\ v(x)=C_{2}x_{N}^{\frac{\alpha_{2}}{2}}+\displaystyle\int_{\mathbb{R}_{+}^{N}}G^{2}_{\infty}(x,y)u^{p_{2}}(y)v^{q_{2}}(y)dy. \end{array} \right. \end{eqnarray} are derived. Based on this result we deal with integral equations (2) instead of (1) and obtain the regularity. Especially, by the method of moving planes in integral forms which is established by Chen-Li-Ou [12], we obtain the nonexistence of positive solutions of integral equations (2) under only local integrability assumptions.
keywords: Kelvin transform. moving planes in integral forms nonexistence integral equations equivalence Fractional laplacians
DCDS-B
Method of sub-super solutions for fractional elliptic equations
Yanqin Fang De Tang
Discrete & Continuous Dynamical Systems - B 2017 doi: 10.3934/dcdsb.2017212
By applying the method of sub-super solutions, we obtain the existence of weak solutions to fractional Laplacian
$\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{s}}u=f(x,u),&\text{in}\ \Omega , \\ u=0,&\text{in}\ {{\mathbb{R}}^{N}}\backslash \Omega , \\\end{array} \right.$
where
$f:\Omega \text{ }\!\!\times\!\!\text{ }\mathbb{R}\to \mathbb{R}$
is a Caratheódory function.
Let
$ν$
be a Radon measure. Based on the existence result in (1), we derive the existence of weak solutions for the semilinear fractional elliptic equation with measure data
$ \left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{s}}u=f(x,u)+\nu ,&\text{in}\ \Omega , \\ u=0,&\text{in}\ {{\mathbb{R}}^{N}}\backslash \Omega , \\\end{array} \right. $
Some results in[7] are extended.
In addition, we generalize some results to systems of fractional Laplacian equations by constructing subsolutions and supersolutions.
keywords: Fractional Laplacian Radon measure Caratheódory function subsolution supersolution

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