Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems

Abstract

We consider the 2 m-th order elliptic boundary value problem L u = f ( x , u ) on a bounded smooth domain Ω ⊂ R N with Dirichlet boundary conditions on ∂ Ω. The operator L is a uniformly elliptic linear operator of order 2 m whose principle part is of the form ( − ∑ i , j = 1 N a i j ( x ) ∂ 2 ∂ x i ∂ x j ) m . We assume that f is superlinear at the origin and satisfies lim s → ∞ f ( x , s ) s q = h ( x ) , lim s → − ∞ f ( x , s ) | s | q = k ( x ) , where h , k ∈ C ( Ω ¯ ) 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.