Classification of solutions to several semi-linear polyharmonic equations and fractional equations

Abstract

We are concerned with the following semi-linear polyharmonic equation with integral constraint \begin{align} \left\{\begin{array}{rl} &(-\Delta)^pu=u^\gamma_+ ~~ \mbox{ in }{\mathbb{R}^n},\\ \nonumber &\int_{\mathbb{R}^n}u_+^{\gamma}dx<+\infty, \end{array}\right. \end{align} where $n>2p$, $p\geq2$ and $p\in\mathbb{Z}$. We obtain for $\gamma\in(1,\frac{n}{n-2p})$ that any nonconstant solution satisfying certain growth at infinity is radial symmetric about some point in $\mathbb{R}^{n}$ and monotone decreasing in the radial direction. In the case $p=2$, the same results are established for more general exponent $\gamma\in(1,\frac{n+4}{n-4})$. For the following fractional equation with integral constraint \begin{equation*} \left\{\begin{array}{rl} &(-\Delta)^sv=v^\gamma_+ ~~ \mbox{ in }{\mathbb{R}^n},~~~~\\ &\int_{\mathbb{R}^n}v_+^{\frac{n(\gamma-1)}{2s}}dx<+\infty,~~~~~ \end{array}\right. \end{equation*} where $s\in(0,1)$, $\gamma \in (1, \frac{n+2s}{n-2s})$ and $n\geq 2$, we also complete the classification of solutions with certain growth at infinity. In addition, observe that the assumptions of the maximum principle named decay at infinity in \cite{chen} can be weakened slightly. Based on this observation, we classify all positive solutions of two semi-linear fractional equations without integral constraint.