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

TL;DR

This paper classifies solutions to semi-linear polyharmonic and fractional equations with integral constraints, establishing symmetry and monotonicity properties for solutions under certain growth conditions, and extends these results to fractional cases.

Contribution

It provides new symmetry classification results for solutions of semi-linear polyharmonic and fractional equations with integral constraints, including cases with more general exponents.

Findings

Solutions are radially symmetric and decreasing under growth conditions

Classification of solutions for fractional equations with integral constraints

Extension of maximum principle assumptions in solution analysis

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.