Symmetry and Nonexistence of Positive Solutions for a Fractional Laplacion System with Coupled Terms

In this paper, we study the problem for a nonlinear elliptic system involving fractional Laplacion: (equation 1.1) where 0 < α, β < 2, p, q > 0 and max{p, q} ≥ 1, α + γ > 0, β + τ > 0, n ≥ 2. First of all, while in the subcritical case, i.e. n + α + γ − p(n − α) − (q + 1)(n − β) > 0, n + β + τ − (p + 1)(n − α) − q(n − β) > 0, we prove the nonexistence of positive solution for the above system in R n . Moreover, though Doubling Lemma to obtain the singularity estimates of the positive solution on bounded domain Ω. In addition, while in the critical case, i.e. n+α+γ −p(n−α)−(q + 1)(n−β) = 0, n+β +τ −(p+ 1)(n−α)−q(n−β) = 0, we show that the positive solution of above system are radical symmetric and decreasing about some point by using the method of Moving planes in Rn Mathematics Subject Classification (2020): 35R11, 35A10, 35B06.

Monotonicity and Symmetry of Solutions to Fractional Laplacian in Strips

In this paper, using the method of moving planes, we study the monotonicity in some directions and symmetry of the Dirichlet problem involving the fractional Laplacian − Δ α / 2 u x = f u x , x ∈ Ω , u x > 0 , x ∈ Ω , u x = 0 , x ∈ ℝ n \ Ω , in a slab-like domain Ω = ℝ n − 1 × 0 , h ⊂ ℝ n .

Radial symmetry for a generalized nonlinear fractional p-Laplacian problem

This paper first introduces a generalized fractional p-Laplacian operator (–Δ)sF;p. By using the direct method of moving planes, with the help of two lemmas, namely decay at infinity and narrow region principle involving the generalized fractional p-Laplacian, we study the monotonicity and radial symmetry of positive solutions of a generalized fractional p-Laplacian equation with negative power. In addition, a similar conclusion is also given for a generalized Hénon-type nonlinear fractional p-Laplacian equation.

Monotonicity results for the fractional p-Laplacian in unbounded domains

In this paper, we develop a direct method of moving planes in unbounded domains for the fractional p-Laplacians, and illustrate how this new method to work for the fractional p-Laplacians. We first proved a monotonicity result for nonlinear equations involving the fractional p-Laplacian in [Formula: see text] without any decay conditions at infinity. Second, we prove De Giorgi conjecture corresponding to the fractional p-Laplacian under some conditions. During these processes, we introduce some new ideas: (i) estimating the singular integrals defining the fractional p-Laplacian along a sequence of approximate maxima; (ii) estimating the lower bound of the solutions by constructing sub-solutions.

Monotonicity and symmetry of positive solutions to degenerate quasilinear elliptic systems in half-spaces and strips

<p style='text-indent:20px;'>By means of the method of moving planes, we study the monotonicity of positive solutions to degenerate quasilinear elliptic systems in half-spaces. We also prove the symmetry of positive solutions to the systems in strips by using similar arguments. Our work extends the main results obtained in [<xref ref-type="bibr" rid="b16">16</xref>,<xref ref-type="bibr" rid="b20">20</xref>] to the system, in which substantial differences with the single cases are presented.</p>

Radial symmetry of nonnegative solutions for nonlinear integral systems

<p style='text-indent:20px;'>In this paper, we investigate the nonnegative solutions of the nonlinear singular integral system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \left\{ \begin{array}{lll} u_i(x) = \int_{\mathbb{R}^n}\frac{1}{|x-y|^{n-\alpha}|y|^{a_i}}f_i(u(y))dy,\quad x\in\mathbb{R}^n,\quad i = 1,2\cdots,m,\\ 0&lt;\alpha&lt;n,\quad u(x) = (u_1(x),\cdots,u_m(x)),\nonumber \end{array}\right. \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;a_i/2&lt;\alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ f_i(u) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 1\leq i\leq m $\end{document}</tex-math></inline-formula>, are real-valued functions, nonnegative and monotone nondecreasing with respect to the independent variables <inline-formula><tex-math id="M4">\begin{document}$ u_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ u_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \cdots $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ u_m $\end{document}</tex-math></inline-formula>. By the method of moving planes in integral forms, we show that the nonnegative solution <inline-formula><tex-math id="M8">\begin{document}$ u = (u_1,u_2,\cdots,u_m) $\end{document}</tex-math></inline-formula> is radially symmetric when <inline-formula><tex-math id="M9">\begin{document}$ f_i $\end{document}</tex-math></inline-formula> satisfies some monotonicity condition.</p>