Nonlinew Anaiysiv, Theory, Methodx & AppIicatiom. Vo1.6. PP. 677-684.1981. Rinted in Gmat Britain 0362-546X/81,06%77-08 SOZ.O&% 83 ,981 Pergainon Press zyxwvutsrqpon Ltd. zyxwvutsr NONRESONANCE AND EXIkTENCE FOR NONLINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS J. MAWHIN ~nk-mit~ Catholique de Louvain, Institut de Mathkmatique, B-1348 Louvain-La-Neuve, Belgium and J. R. WARD, JR. Department of Mathematics, University of Alabama, University, Alabama 35486, U.S.A. (Received 24 June 1980; in revisedform 26 September 1980) Key words andphrases: Elliptic boundary value problems, nonresonance, nonlinear. LET A‘2 BE a bounded open set in R", n 3 1. Let zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML L = 1 (- l)“‘D’(Ui/X)D’) Ii/. Ii1 bn be a uniformly strongly elliptic operator acting on functions defined on n. We also assume that a . are real valued functions in L"(Q) for Ii/, /jl < m, with aii uniformly continuous on R for ;fjzlilj’ = m. Here zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA i zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA and j denote multiindices with Ii1= i, + i, + . . . + in for i = (iI, i,, . . . , in). The reader is referred to [I], [S], or [4] for the results on elliptic operators used herein. In this note we give a non-resonance condition for the existence of a weak solution to the Dirichlet problem Lu = g(x,Du)u + h(x,Du), (1.1) aS4/an = 0 (i = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH 0, 1, . . . , m - l)onaR, where a/an is the outward normal derivative on an, and Du denotes that vector consisting of u and all of its derivatives up to order m. Let A, < A, < . . . denote the eigenvalues of the linear homogeneous problem Lu= All, (1.2) &lad = 0 (OgiGm-I)on&2. Landesman and Lazer [7] showed that (1.1) has a solution provided g and h are continuous bounded functions and for some integer N 2 1 there are numbers p and 4 such that AN< p < g(x, Du)< q < AN+ 1 for all (x, Du). (1.3) They actually consider only the case m = 1, but their proof does not really depend on that. Condition (1.3) is a nonresonance condition in the sense that if the linearized version of (1.1) Lu = p(x)u + h(x) (1.4) satisfies (1.3) then it has a unique weak solution for any h E E(n) and there is a constant k > 0