Kavli Affiliate: Feng Yuan
| First 5 Authors: Zhuoran Du, Zhenping Feng, Yuan Li, ,
| Summary:
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 $n>2p$, $pgeq2$ and $pinmathbb{Z}$. We obtain for
$gammain(1,frac{n}{n-2p})$ 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
$gammain(1,frac{n+4}{n-4})$. 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 $sin(0,1)$, $gamma in (1,
frac{n+2s}{n-2s})$ and $ngeq 2$, 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.
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