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--- 分析与偏微分方程讨论班(2023春季第8讲)

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

杜卓然

(湖南大学)

时间：2023-06-05  10：00-11：00  (周一上午)  

地点: 腾讯会议 ID：475-997-573

    腾讯会议链接：https://meeting.tencent.com/dm/niX43xdH15oi

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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