Nonlinear Analysis. Theory. Methods & Applicotions,Vol 3 No 2, pp. 157-174 6 Pergamon Press Ltd. 1979 Printed in Great Britain. 0362.546X:79/03(11-0157 SO2 00/O zyxwvuts SOME NONLINEAR ELLIPTIC PROBLEM S WITH ASYMPTOTIC CONDITIONS* VIERI BENCI Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, U.S.A., and Istituto di Matematiche Applicate “U. Dini”, University of Pisa, Piss, Italy and DONATO FORTUNATE Istituto di Matematica Applicata, University of Bari, Via Re David 200, 70125 Bari, Italy zyxwvutsrqponmlkjih (Received 30 .hnuury 1978) Key words: Elliptic operators, Sobolov spaces, variational inequalities, maximum principle. INTRODUCTION IN ELLIPTIC problems in unbounded domains the boundary conditions include the asymptotic behavior of the solutions as (xl approaches infinity. Thus, it is of primary importance to know which asymptotic conditions are reasonable to require for the solutions of such problems. The aims of this paper are to investigate how the asymptotic conditions depend on the diffe- rential equation considered, and to give some existence and “non-existence” theorems for such a kind of problems. Consider e.g., the following problem: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC -Au + f(u) = g UJ;* = 0 (0.1) where lim f(t) = f co ; g E C:(n) and R is an unbounded domain in zyxwvutsrqponmlkjihgfe R” with smooth boundi;y*(zote that if 52 is bounded, problem (0.1) always admits a solution u E C’(Q); cfr. e.g. L zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 21). In order to apply the techniques of nonlinear analysis to problem (O.l), one must specify spaces of functions satisfying the desired asymptotic conditions. Thus it seems to be a natural choice for such kind of problems to use Sobolev spaces because the Hiilder spaces do not include the asymptotic conditions. However, in general, the ordinary Sobolev spaces are a too narrow class in order to try to find solutions to problems of type (0.1); this is shown in a counterexample given in Section 4. For this reason we will study problems ofthis kind in weighted Sobolev spaces in the framework of the theory developed by the authors [3,4]. *This research was partially supported by the Consiglio Nazionale delle Ricerche 157