An approach to nonlinear elliptic boundary value problems

Abstract

AbstractLet D ⊂ Rnbe a bounded domain andL: domL⊂L2(D) →L2(D) be a self-adjoint operator of finite dimensional kernel. Letf:D×R→Rbe a function satisfying the Carathéodory condition. Assume that there are constants λ > 0 and δ ∈ [0, 1] such thatand that.Then with the aid of a generalized Krasnosel'skii's theorem it has been proved that under conditions exactly analogous to those of Landesman and Lazer there existsu∈L2(D) such thatL(u)(x) = f(x, u(x))for∀x∈D. This result is then used to prove the existence of weak solutions of nonlinesr elliptic boundary value problems.Other abstract results applicable to ordinary and partial differential equations have also been proved.