PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 3, Pages 755–765 S 0002-9939(99)05485-4 Article electronically published on September 9, 1999 RESONANCE PROBLEMS FOR THE ONE-DIMENSIONAL p-LAPLACIAN PAVEL DR ´ ABEK AND STEPHEN B. ROBINSON (Communicated by Hal L. Smith) Abstract. We consider resonance problems for the one dimensional p-La- placian, and prove the existence of solutions assuming a standard Landesman- Lazer condition. Our proofs use variational techniques to characterize the eigenvalues, and then to establish the solvability of the given boundary value problem. 1. Introduction Consider the boundary value problem −(|u ′ | p−2 u ′ ) ′ − λ n |u| p−2 u − f (u)+ h = 0 in (0, 1), u(0) = u(1) = 0, (1) where f : R → R is a bounded continuous function such that the limits lim t→±∞ f (t)= f (±∞) exist, p> 1, h ∈ L q (0, 1) such that p + q = pq, and λ n is an eigenvalue of the associated homogeneous problem −(|u ′ | p−2 u ′ ) ′ − λ|u| p−2 u = 0 in (0, 1), u(0) = u(1) = 0. (2) It is known that the eigenvalues of (2) are simple, positive, and form an un- bounded increasing sequence, {λ n }, whose eigenspaces are spanned by functions {φ n (x)}⊂ W 1,p 0 (0, 1) ∩ C 1 [0, 1] such that φ n has n − 1 evenly spaced zeros in (0, 1), ||φ n || L p = 1, and φ ′ n (0) > 0. See [6], pages 174-183, for further details and references. We will show that (1) is solvable if either f (+∞)  1 0 φ + n + f (−∞)  1 0 φ − n >  1 0 φ n h>f (+∞)  1 0 φ − n + f (−∞)  1 0 φ + n (3) Received by the editors April 21, 1998. 2000 Mathematics Subject Classification. Primary 34B15. The first author’s research was sponsored by the Grant Agency of the Czech Republic, Project no. 201/97/0395, and partly by the Ministery of Education of the Czech Republic, Project no. VS97156. c 1999 American Mathematical Society 755 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use