Existence of positive solutions for some polyharmonic nonlinear boundary-value problems

Abstract

<p>We present existence results for the polyharmonic nonlinear elliptic boundary-value problem</p> <pre>(-Δ)^m u = ƒ(·, u) in B<br> (∂/ ∂v)^j u = 0 on ∂B, 0 ≤ j ≤ m - 1.</pre> <p>(in the sense of distributions), where B is the unit ball in ℝⁿ and n ≥ 2. The nonlinearity ƒ(x, t) satisfies appropriate conditions related to a Kato class of functions K_m,n. Our approach is based on estimates for the polyharmonic Green function with zero Dirichlet boundary conditions and on the Schauder fixed point theorem.</p>