Nonlinror Analwis. Theory. Merhodr & Applicorions. Vol. 3. No 5. pp 629-634 0 Pergamon Press Ltd 1479. Prmted in Great Brmin 0362-546X 79!0901-0629 102.00’0 zyxwvuts PERTURBATIONS OF SECOND ORDER LINEAR ELLIPTIC PROBLEM S BY NONLINEARITIES WITHOUT LANDESMAN-LAZER CONDITION zyxwvutsrqponmlkjihgfedcbaZY DJAIRO GUEDES DE FIGUEIREDO Universidade de Brasilia, Department0 de MatemBtica, Brasilia. D.F., Brazil and WEI-MING NI Courant Institute of Mathematical Sciences (Recehvd 30 Auyusr 1978) Key Words: Nonlinear elliptic problems, maximum principle, LI priori bounds, Leray-Schauder degree theory. LET P’ be a second order symmetric uniformly elliptic operator with smooth coefficients acting on real valued functions defined in a bounded smooth domain R in RN. Let us denote by ;1i the first eigenvalue of the eigenvalue problem YU = lu in R and u = 0 on 80. It is well known [l] that ii is a simple eigenvalue and that there is a corresponding smooth eigenfunction 4 > 0 in R. It follows from the Hopf-Giraud maximum principle [2, p. 1511 that the outward normal derivative &#J/& is ~0 on the boundary. In this paper we consider the Dirichlet problem _YfJ - Itl.4 + g(u) = h in Q, u = Oon&$ (1) where g: R + R is a bounded continuous function. An intensive research in this problem started after the 1970 paper [3] of Landesman and Lazer. Their result is essentially as follows. Problem (1) has a solution 1.4 E zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Wi* ‘(Cl) n W2’ ‘(Cl) for every h E c(CY2) provided where g* = lim g(u). (Wm,p(Q) is the Sobolev space of LP-functions whose derivatives in the u-+a distribution sense are in E’(Q), see, for example, Adams [4]). In fact they do not restrict to the first eigenvalue, but they suppose that g_ d g(u) d g+, for all u E R. The hypothesis on the simplicity of the eigenvalue, the condition g_ d g(u) Q g+, the order of the elliptic operator and even the condition on the boundedness of g have all been relaxed by several authors in recent years. However in all cases a condition like (2) has been assumed. Recently FuEik and Krbec [5] and Hess [6] have been able to prove results on the existence of solutions of (1) for h orthogonal to C$in L? in the case that g+ = g_ = 0. But in every case there are hypotheses on the speed that g goes to 0 at co. Namely they assume that g is odd and