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

Abstract

In this paper, we present a new sufficient condition to get a priori L∞-estimates for positive solutions of higher-order elliptic equations in a smooth bounded convex domain of RN with Navier boundary conditions or for radially symmetric solutions in the ball with Dirichlet boundary conditions. A priori L∞-estimates for positive solutions of the second-order elliptic system in a smooth bounded convex domain of RN 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: liminfs→+∞f(s)s>Λ1, limsups→0f(s)s<Λ1, where Λ1 is the first eigenvalue of (−Δ)m with Navier boundary conditions and the weak subcritical growth condition: lims→+∞⁡f(s)sσ=0, where σ=N+2mN−2m.