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【学术报告及分析与偏微分方程讨论班(2023春第8讲)】Classification of solutions to semi-linear polyharmonic equations and fractional equations
发布日期:2023-06-01    点击:
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--- 分析与偏微分方程讨论班(2023季第8)

 

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


杜卓然

(湖南大学)


时间:2023-06-05  1000-1100  (一上)  


地点: 腾讯会议 ID475-997-573

    腾讯会议链接:https://meeting.tencent.com/dm/niX43xdh75oi


摘要: We study the following semi-linear polyharmonic equation with integral constraint

\begin{eqnarray}

\left\{\begin{array}{rl}

&(-\Delta)^pu=u^\gamma_+   \mbox{ in }{\mathbb{R}^n},\\

&\int_{\mathbb{R}^n}u_+^{\gamma}dx<+\infty,

\end{array}\right.

\end{eqnarray}

where  $n>2p$, $p\geq2$ and $p\in\mathbb{Z}$. We obtain for $\gamma\in(1,\frac{n+2p}{n-2p})$ that any  nonconstant solution   satisfying certain conditions at infinity is radial symmetric about some point in $\mathbb{R}^{n}$  and monotone decreasing in the radial direction. For the following fractional equation with integral constraint

\begin{eqnarray}

\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{eqnarray}

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.


报告人简介: 杜卓然,湖南大学数学学院副教授。研究领域为偏微分方程理论与非线性分析。在Adv. Math., Calc. Var. PDE, JDE等权威期刊上发表论文多篇。

 

邀请人:戴蔚

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