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Abstract

In this paper, we are concerned with the thirdorder equations
$$\left\{\begin{array}{cc}{\left(\Delta \right)}^{\frac{3}{2}}u={u}^{\frac{d+3}{d3}},\phantom{\rule{1em}{0ex}}\hfill & x\in {\mathbb{R}}^{d},\hfill \\ u\in {C}^{3}\left({\mathbb{R}}^{d}\right),\phantom{\rule{1em}{0ex}}\hfill & u\left(x\right)>0,\phantom{\rule{2.77626pt}{0ex}}x\in {\mathbb{R}}^{d},\hfill \end{array}\right.$$
and
$$\left\{\begin{array}{cc}{\left(\Delta \right)}^{\frac{3}{2}}u=\left(\frac{1}{x{}^{6}}\ast u{}^{2}\right)u,\phantom{\rule{1em}{0ex}}\hfill & x\in {\mathbb{R}}^{d},\hfill \\ u\in {C}^{3}\left({\mathbb{R}}^{d}\right),\phantom{\rule{1em}{0ex}}\hfill & u\left(x\right)>0,\phantom{\rule{2.77626pt}{0ex}}x\in {\mathbb{R}}^{d},\phantom{\rule{0.3em}{0ex}}d\ge 7,\hfill \end{array}\right.$$
with
${\u1e22}^{\frac{3}{2}}$critical
nonlinearity. By showing the equivalence between the PDEs and the
corresponding integral equations and using results from Chen et al.
(2006) and Dai et al. (2018), we prove that positive classical solutions
$u$
to the above equations are radially symmetric about some point
${x}_{0}\in {\mathbb{R}}^{d}$ and derive the
explicit forms for
$u$.

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Keywords
fractional Laplacians, odd order, positive smooth
solutions, radial symmetry, uniqueness, equivalence

Mathematical Subject Classification 2010
Primary: 35R11
Secondary: 35B06, 35J91

Milestones
Received: 21 July 2017
Revised: 31 January 2018
Accepted: 3 February 2018
Published: 11 April 2018

