Vol. 295, No. 2, 2018

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Classification of positive smooth solutions to third-order PDEs involving fractional Laplacians

Wei Dai and Guolin Qin

Vol. 295 (2018), No. 2, 367–383
Abstract

In this paper, we are concerned with the third-order equations

$\left\{\begin{array}{cc}{\left(-\Delta \right)}^{\frac{3}{2}}u={u}^{\frac{d+3}{d-3}},\phantom{\rule{1em}{0ex}}\hfill & x\in {ℝ}^{d},\hfill \\ u\in {C}^{3}\left({ℝ}^{d}\right),\phantom{\rule{1em}{0ex}}\hfill & u\left(x\right)>0,\phantom{\rule{2.77626pt}{0ex}}x\in {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 {ℝ}^{d},\hfill \\ u\in {C}^{3}\left({ℝ}^{d}\right),\phantom{\rule{1em}{0ex}}\hfill & u\left(x\right)>0,\phantom{\rule{2.77626pt}{0ex}}x\in {ℝ}^{d},\phantom{\rule{0.3em}{0ex}}d\ge 7,\hfill \end{array}\right\$

with ${Ḣ}^{\frac{3}{2}}$-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 ${x}_{0}\in {ℝ}^{d}$ and derive the explicit forms for $u$.

Keywords
fractional Laplacians, odd order, positive smooth solutions, radial symmetry, uniqueness, equivalence
Mathematical Subject Classification 2010
Primary: 35R11
Secondary: 35B06, 35J91