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

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain Omega subset of R(N) with Dirichlet boundary conditions u = partial derivative/partial derivative nu u = ... = (partial derivative/partial derivative nu)(m-1) u = 0 on partial derivative Omega. The operator L is a uniformly elliptic operator of order 2m given by L = (-Sigma(N)(i,j=1) a(ij)(x)partial derivative(2)/partial derivative x(i)partial derivative x(j))(m) + Sigma vertical bar alpha vertical bar <= 2m-1 b(alpha)(x)D(alpha.) For the nonlinearity we assume that lim(s ->infinity) f(x,s)/s(q) = h(x), lim(s ->-infinity) f(x,s)/vertical bar s vertical bar(q) = k(x) where h, k is an element of C ((Omega) over bar) are positive functions and q > 1 if N <= 2m, 1 < q < N+2m/N-2m if N > 2m. We prove a priori bounds, i. e, we show that parallel to u parallel to (L infinity) ((Omega)) <= C for every solution u, where C > 0 is a constant. 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 R(+)(N) with Dirichlet boundary conditions on. partial derivative R(+)(N) and q > 1 if N <= 2m, 1 < q <= N+2m/N-2m if N > 2m then u = 0.