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

TL;DR

This paper establishes a priori bounds for solutions to higher order elliptic boundary value problems in bounded domains, using a new Liouville theorem on a half-space to control solution behavior and ensure boundedness.

Contribution

It introduces a novel Liouville-type theorem for higher order elliptic equations on half-spaces, enabling the derivation of a priori bounds for solutions in bounded domains.

Findings

Proved a Liouville theorem for higher order elliptic equations on half-spaces.

Established a priori bounds for solutions of higher order elliptic problems.

Demonstrated the boundedness of solutions regardless of sign-changing behavior.

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.