A direct method of moving planes for a fully nonlinear nonlocal system
Pengyan Wang Pengcheng Niu
Communications on Pure & Applied Analysis 2017, 16(5): 1707-1718 doi: 10.3934/cpaa.2017082
In this paper we consider the system involving fully nonlinear nonlocal operators:
$ \left\{\begin{array}{ll}{\mathcal F}_{α}(u(x)) = C_{n,α} PV ∈t_{\mathbb{R}^n} \frac{F(u(x)-u(y))}{|x-y|^{n+α}} dy=v^p(x)+k_1(x)u^r(x),\\{\mathcal G}_{β}(v(x)) = C_{n,β} PV ∈t_{\mathbb{R}^n} \frac{G(v(x)-v(y))}{|x-y|^{n+β}} dy=u^q(x)+k_2(x)v^s(x),\end{array}\right.$
$0<α, β<2, $
$p, q, r, s>1, $
$k_1(x), k_2(x)\geq0.$
A narrow region principle and a decay at infinity are established for carrying on the method of moving planes. Then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space. Furthermore non-existence of positive solutions to the system on a half space is derived.
keywords: Fully nonlinear nonlocal operator narrow region principle decay at infinity method of moving planes non-existence
Liouville's theorem for a fractional elliptic system
Pengyan Wang Pengcheng Niu
Discrete & Continuous Dynamical Systems - A 2019, 39(3): 1545-1558 doi: 10.3934/dcds.2019067
In this paper, we investigate the following fractional elliptic system
$\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{\alpha /2}}u(x) = f(x){{v}^{q}}(x),&x\in {{R}^{n}}, \\ {{(-\Delta )}^{\beta /2}}v(x) = h(x){{u}^{p}}(x),&x\in {{R}^{n}}, \\\end{array} \right.$
where $1≤p, q < ∞$, $0 < α, β < 2$, $f(x)$ and $h(x)$ satisfy suitable conditions. Applying the method of moving planes, we prove monotonicity without any decay assumption at infinity. Furthermore, if $ α = β$, a Liouville theorem is established.
keywords: The fractional Laplace system Liouville's theorem method of moving planes
Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations
Leyun Wu Pengcheng Niu
Discrete & Continuous Dynamical Systems - A 2019, 39(3): 1573-1583 doi: 10.3934/dcds.2019069
In this paper, we consider the fractional p-Laplacian equation
$( - \Delta )_p^su(x) = f(u(x)), $
where the fractional p-Laplacian is of the form
$( - \Delta )_p^su(x) = {C_{n, s, p}}PV\int_{{\mathbb{R}^n}} {\frac{{{{\left| {u(x) - u(y)} \right|}^{p - 2}}(u(x) - u(y))}}{{{{\left| {x - y} \right|}^{n + sp}}}}} dy.$
By proving a narrow region principle to the equation above and extending the direct method of moving planes used in fractional Laplacian equations, we establish the radial symmetry in the unit ball and nonexistence on the half space for the solutions, respectively.
keywords: Fractional p-Laplacian equation narrow region principle direct method of moving planes radial symmetry nonexistence
A Liouville type theorem to an extension problem relating to the Heisenberg group
Xinjing Wang Pengcheng Niu Xuewei Cui
Communications on Pure & Applied Analysis 2018, 17(6): 2379-2394 doi: 10.3934/cpaa.2018113

We establish a Liouville type theorem for nonnegative cylindrical solutions to the extension problem corresponding to a fractional CR covariant equation on the Heisenberg group by using the generalized CR inversion and the moving plane method.

keywords: Heisenberg group extension problem Liouville type theorem generalized CR inversion moving plane method

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