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

Abstract

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega$ 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 $(-\Delta {)}^{m}u=u^q$ in a half-space with Dirichlet boundary conditions and if $q>1$ is subcritical then $u$ vanishes identically.

23 pages