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

Wei Dai and Guolin Qin

Vol. 295 (2018), No. 2, 367–383

DOI: 10.2140/pjm.2018.295.367

Abstract

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

$$\left\{\begin{array}{cc}{\left(-\Delta \right)}^{\frac{3}{2}}u=u^{\frac{d+3}{d-3}},\hfill & x\in {ℝ}^d,\hfill \\ u\in C^3\left({ℝ}^d\right),\hfill & u\left(x\right)>0,x\in {ℝ}^d,\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{cc}{\left(-\Delta \right)}^{\frac{3}{2}}u=\left(\frac{1}{|x{|}^6}\ast |u{|}^2\right)u,\hfill & x\in {ℝ}^d,\hfill \\ u\in C^3\left({ℝ}^d\right),\hfill & u\left(x\right)>0,x\in {ℝ}^d,d\ge 7,\hfill \end{array}\right.$$

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

Milestones

Received: 21 July 2017

Revised: 31 January 2018

Accepted: 3 February 2018

Published: 11 April 2018