Liouville type theorems for poly-harmonic Navier problems
Liouville type theorems for poly-harmonic Navier problems
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DOI：
10.3934/dcds.2013.33.3937
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发表时间：
2013-03
影响因子：
1.1
通讯作者：
Linfen Cao;Wenxiong Chen
中科院分区：
文献类型：
--
作者：
Linfen Cao;Wenxiong Chen
本文考虑半空间R^n+$上具有Navier边界条件的半线性多调和方程：\Begin{方程}\Left\Begin{数组}{L}(-\三角形)^{\FRAC{\α}{2}}u=u^p，\\\：\：\\Mbox{in}\，\R^n_+，\\u=-\三角形u=\cdots=(-\triangle)^{\frac{\alpha}{2}-1}u=0，\\\Mbox{开}\\部分R^n_+，\end{数组}\右。\LABEL{phe1}\end{公式}，其中$\Alpha$是$0$和$n$之间的任意偶数，$p>；1$。首先，我们证明了(1)在很温和的增长条件下等价于如下积分方程u(X)=IntR^n+}G(x，y，α)u^p(Y)dy，R^n+，Label{ie0}\end{方程}，其中$G(x，y，α)$是半空间上对应于相同Navier边界条件的格林函数.然后，将积分形式的移动平面方法与一类Kelvin变换相结合，仅在局部可积条件下，得到了积分方程(2)在亚临界和临界情况下正解的不存在性。这大大削弱了文献[3]中关于解的全局可积性的假设。我们关于积分方程(2)的结果对在$0$和$n$之间的所有实值$\α$都是有效的。最后，我们建立了偏微分方程解的Liouvile型定理，大大削弱了了解的增长条件，从而推广了郭和刘的结果[21]。
In this paper we consider the following semi-linear poly-harmonic equation with Navier boundary conditions on the half space $R^n_+$: \begin{equation} \left\{\begin{array}{l} (-\triangle)^{\frac{\alpha}{2}} u=u^p,\ \ \ \ \ \:\:\: \:\:\:\:\:\ \:\:\ \ \ \ \ \ \ \ \ \ \ \ \:\:\:\:\ \mbox{in}\,\ R^n_+,\\ u=-\triangle u=\cdots=(-\triangle)^{\frac{\alpha}{2}-1}u=0, \ \ \ \mbox{on}\ \partial R^n_+, \end{array} \right. \label{phe1} \end{equation} where $\alpha$ is any even number between $0$ and $n$, and $p>1$. First we prove that (1) is equivalent to the following integral equation \begin{equation} u(x)=\int_{R^n_+}G(x,y,\alpha) u^p(y)dy,\,\,\,\,\, x\in\,R^n_+, \label{ie0} \end{equation} under some very mild growth condition, where $G(x, y,\alpha)$ is the Green's function associated with the same Navier boundary conditions on the half-space . Then by combining the method of moving planes in integral forms with a certain type of Kelvin transform, we obtain the non-existence of positive solutions for integral equation (2) in both subcritical and critical cases under only local integrability conditions. This remarkably weaken the global integrability assumptions on solutions in paper [3]. Our results on integral equation (2) are valid for all real values $\alpha$ between $0$ and $n$. Finally, we establish a Liouville type theorem for PDE (1), and this generalizes Guo and Liu's result [21] by significantly weaken the growth conditions on the solutions.