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
$u$.
