Existence of solutions to nonlinear, subcritical 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 $\Omega \subset {\mathbb{R}}^N$ with Dirichlet boundary conditions on $\partial \Omega$. The operator $L$ is a uniformly elliptic linear operator of order $2m$ whose principle part is of the form $(-{\sum }_{i,j=1}^N{a}_{ij}(x)\frac{{\partial }^2}{\partial x_i\partial x_j}{)}^m$. We assume that $f$ is superlinear at the origin and satisfies $\underset{s\to \infty }{\lim }\frac{f(x,s)}{s^q}=h(x)$, $\underset{s\to -\infty }{\lim }\frac{f(x,s)}{|s{|}^q}=k(x)$, where $h,k\in C(\overline{\Omega })$ are positive functions and $q>1$ is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution.