A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

Authors:	Wolfgang Reichel Tobias Weth

Institution:	1.Institut für Analysis,Universit?t Karlsruhe,Karlsruhe,Germany;2.Mathematisches Institut,Universit?t Giessen,Giessen,Germany

Abstract:	We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain ${\Omega\subset\mathbb R^N}$ with Dirichlet boundary conditions ${u= \frac{\partial}{\partial\nu}u= \cdots=(\frac{\partial}{\partial\nu})^{m-1} u =0}$ on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by ${L=\left(-\sum_{i,j=1}^N a_{ij}(x) \frac{\partial^2}{\partial x_i\partial x_j} \right)^m +\sum_{\|\alpha\|\leq 2m-1} b_\alpha(x) D^\alpha}$ . For the nonlinearity we assume that ${{\rm lim}_{s\to\infty}\frac{f(x,s)}{s^q}=h(x)}$ , ${{\rm lim}_{s\to-\infty}\frac{f(x,s)}{\|s\|^q}=k(x)}$ where ${h,k\in C(\overline{\Omega})}$ are positive functions and q > 1 if N ≤ 2m, ${1 < q < \frac{N+2m}{N-2m}}$ if N > 2m. We prove a priori bounds, i.e, we show that ${\\|u\\|_{L^\infty(\Omega)} \leq C}$ for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ)^m u = u ^q in ${\mathbb R^N_+}$ with Dirichlet boundary conditions on ${\partial\mathbb R^N_+}$ and q > 1 if N ≤ 2m, ${1 < q\leq\frac{N+2m}{N-2m}}$ if N > 2m then ${u\equiv 0}$ .

Keywords:	Higher order equation A priori bounds Liouville theorems Moving plane method
本文献已被 SpringerLink 等数据库收录！

设为首页 | 免责声明 | 关于勤云 | 加入收藏