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

By Wolfgang Reichel, Tobias Weth. 2007

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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)mu  =  uq 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 $${u\equiv 0}$$.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/3e14d30b8d09951e5137955e0551d259a4b607e1