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 Ω ⊂ R with Dirichlet boundary conditions u = ∂ ∂ν u = . . . = ( ∂ ∂ν )u = 0 on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by L = ( − ∑N i,j=1 aij(x) ∂ ∂xi∂xj )m + ∑ |α|≤2m−1 bα(x)D . For the nonlinearity we assume that lims→∞ f(x,s) sq = h(x), lims→−∞ f(x,s) |s|q = k(x) where h, k ∈ C(Ω) 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 ‖u‖L∞(Ω) ≤ 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 (−∆)mu = u in R+ with Dirichlet boundary conditions on ∂R+ and q > 1 if N ≤ 2m, 1 < q ≤ N+2m N−2m if N > 2m then u ≡ 0.