论文标题
将溶液分类为几个半线性多谐和方程和分数方程
Classification of solutions to several semi-linear polyharmonic equations and fractional equations
论文作者
论文摘要
我们关注以下半线性多谐和方程，并具有积分约束\ begin {align} \ left \ {\ begin {array} {rl} {rl}＆（ - δ） ＆\ int _ {\ mathbb {r}^n} u _+^γdx<+\ infty，\ end {array} \ right。 \ end {align}其中$ n> 2p $，$ p \ geq2 $和$ p \ in \ mathbb {z} $。我们以$γ\在（1，\ frac {n} {n-2p}）$中获得，任何满足Infinity在Infinity生长时的非稳定解决方案均在$ \ Mathbb {r}^{n} $中的某个点上是径向对称的，并且沿径向方向降低了单调。在情况下，$ p = 2 $，（1，\ frac {n+4} {n-4}）$为更通用的指数$γ\建立了相同的结果。对于以下具有积分约束\ begin {qore*} \ left \ {\ {arnay} {arl} {rl}＆（ - δ）^sv = v^γ_+ ~~ \ mbox {in} ＆\ int _ {\ mathbb {r}^n} v _+^{\ frac {n（γ-1）} {2S}} dx <+\ infty，~~~~~~~~~~~~~~~~~ end {array} \ right。 \ end {equation*}其中$ s \ in（0,1）$，$γ\ in（1，\ frac {n+2s} {n-2s} {n-2s}）$和$ n \ geq 2 $，我们还完成了与Infinity某些增长的解决方案分类。此外，观察到，在\ cite {chen}中命名为无穷大的最大原理衰减的假设可以稍微削弱。基于此观察结果，我们将两个半线性分数方程的所有正溶液分类，而没有积分约束。
We are concerned with the following semi-linear polyharmonic equation with integral constraint \begin{align} \left\{\begin{array}{rl} &(-Δ)^pu=u^γ_+ ~~ \mbox{ in }{\mathbb{R}^n},\\ \nonumber &\int_{\mathbb{R}^n}u_+^γdx<+\infty, \end{array}\right. \end{align} where $n>2p$, $p\geq2$ and $p\in\mathbb{Z}$. We obtain for $γ\in(1,\frac{n}{n-2p})$ 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 $γ\in(1,\frac{n+4}{n-4})$. For the following fractional equation with integral constraint \begin{equation*} \left\{\begin{array}{rl} &(-Δ)^sv=v^γ_+ ~~ \mbox{ in }{\mathbb{R}^n},~~~~\\ &\int_{\mathbb{R}^n}v_+^{\frac{n(γ-1)}{2s}}dx<+\infty,~~~~~ \end{array}\right. \end{equation*} where $s\in(0,1)$, $γ\in (1, \frac{n+2s}{n-2s})$ 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.
