Existence of Solutions to Nonlinear, Subcritical Higher-order Elliptic Dirichlet Problems

Reichel, Wolfgang
Weth, Tobias

facul_09; 12

Date: 16. 6. 2009
Abstract:
We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega \subset \R^N$ 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 $( -\sum_{i,j=1}^{N)} a_{ij} (x) \frac{\partial^2}{\partial_x, \partial_{x_j} )^m$. We assume that $f$ is superlinear at the origin and satisfies $\lim_{s \to \infty} \frac{f(x,s)}{s^q} = h(x), \lim_{s \to - \infty} \frac{f(x,s)}{|s|^q} = k(x)$, where $h,k \in C (\bar{\Omega}}$ 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.

Primary MSC: 35J40 Boundary value problems for higher­order, elliptic equations, 35B45 A priori estimates
Keywords: Higher order equation, existence, topological degree, Liouville theorems