Symmetry of positive solutions for equations involving higher-order fractional Laplacian (Q2817008)

This paper deals with problems associated to higher-order linear fractional Laplacian as: NEWLINE\[NEWLINE\begin{aligned} (-\Delta)^s u= 0\quad &\text{in }\mathbb{R}^n_+,\qquad i= 0,1,2,\dots,\\ (-\Delta)^i u= 0\quad &\text{for }x\not\in\mathbb{R}^n_+,\qquad s=m+{\alpha\over 2},\;m\in\mathbb{N},\;0<\alpha<2,\end{aligned}\tag{1}NEWLINE\]NEWLINE the nonlinear higher-order fractional Laplacian NEWLINE\[NEWLINE\begin{aligned} (-\Delta)^s u= u^n\quad &\text{in }\mathbb{R}^n_+,\\ (-\Delta)^i u= 0\quad &\text{for }x\not\in\mathbb{R}^n_+,\quad p>1\end{aligned}\tag{2}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE(-\Delta)^{{\alpha\over 2}} u= f(x_n)> 0,\quad x\in\mathbb{R}^n_+.\tag{3}NEWLINE\]NEWLINE The authors establish the equivalence between (2) and its corresponding integral equation for positive \(u\). Through the method of moving planes, see [\textit{H. Berestycki} and \textit{L. Nirenberg}, Bol. Soc. Bras. Mat., Nova Sér. 22, No. 1, 1--37 (1991; Zbl 0784.35025)], they derive rotational symmetry of positive solutions \(u\) of (1), (3) and show their dependence on the \(x_n\) variable only.