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

2008; Springer Science+Business Media; Volume: 261; Issue: 4 Linguagem: Inglês

10.1007/s00209-008-0352-3

1432-1823

Wolfgang Reichel, Tobias Weth,

Differential Equations and Boundary Problems

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 $${u\equiv 0}$$ .