A priori estimates and existence of positive solutions for higher-order elliptic equations

In this paper, we present a new sufficient condition to get a priori L-infinity-estimates for positive solutions of higher-order elliptic equations in a smooth bounded convex domain of R-N with Navier boundary conditions or for radially symmetric solutions in the ball with Dirichlet boundary conditions. A priori L-infinity-estimates for positive solutions of the second-order elliptic system in a smooth bounded convex domain of R-N with Dirichlet boundary conditions are also established. As usual, these a priori bounds allow us to obtain existence results. Also, by truncation technique combined with minimax method, we obtain existence of positive solution for higher-order elliptic equations of the form (1.1) below when we only assume that the nonlinearity is a nondecreasing positive function satisfying: lim inf(s ->+infinity) f(s)/s > Lambda(1), lim sup(s -> 0) f(s)/s < Lambda(1), where Lambda(1) is the first eigenvalue of (-Delta)(m) with Navier boundary conditions and the weak subcritical growth condition: lim(s ->+infinity) f(s)/s = 0, where sigma = N+2m/N-2m. (C) 2015 Elsevier Inc. All rights reserved.