A priori bounds and a Liouville theorem on a half-space for higher order elliptic Dirichlet problems
| dc.creator | Reichel, Wolfgang | |
| dc.creator | Weth, Tobias | |
| dc.date | 2007-09-18 | |
| dc.date.accessioned | 2026-07-07T08:30:27Z | |
| dc.date.available | 2026-07-07T08:30:27Z | |
| dc.description | We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $Ω$ in $R^N$ with Dirichlet boundary conditions. The operator $L$ is a uniformly elliptic operator of order $2m$. We assume that for $s\to \pm\infty$ the nonlinearity $f(x,s)$ behaves like $|s|^q$ multiplied by a continuous and positive function of $x$. Here the exponent $q$ is subcritical, i.e., $q>1$ if $N<=2m$, $1<q<\frac{N+2m}{N-2m}$ if $N>2m$. We prove a priori bounds, i.e, we show that the $L^\infty$-norm of every solution $u$ is bounded by a constant independent of $u$. 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 a half-space with Dirichlet boundary conditions and if $q>1$ is subcritical then $u$ vanishes identically. | |
| dc.description | 23 pages | |
| dc.identifier | https://arxiv.org/abs/0709.2821 | |
| dc.identifier | http://arxiv.org/abs/0709.2821 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/138248 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35J40; 35B45 | |
| dc.title | A priori bounds and a Liouville theorem on a half-space for higher order elliptic Dirichlet problems | |
| dc.type | text |