Calc. Var. 4, 121-137 (1996) 9 Springer-Verlag 1996 Local mountain passes for semilinear elliptic problems in unbounded domains Manuel del Pino 1,*, Patricio L. Felmer 2,** 1 Department of Mathematics, University of Chicago, Chicago, IL 60637, USA; e-mail: DELPINO@MATH.UCHICAGO.EDU 2 Departamento de Ingenierfa Matem~itica F.C.F.M., Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile; e-mail: PFELMER@DIM.UCHILE.CL Received August 20, 1994 / Accepted February 10, 1995 0 Introduction This paper has been motivated by some works appeared in recent years concern- ing the nonlinear SchrOdinger equation h2 ih - 2mAffj+V(z)~b-"/t~lP-l~b. (0.1) We are interested in solutions of the form ~b(z, t) = exp(-iEt/h)v(z) which are called standing waves. We observe that this ~# satisfies (0.1) if and only if the function v(z) solves the elliptic equation h2 ~m Av -- (V(z) - E)v + 7 I v ]e-1 v = O. (0.2) In [3], Floer and Weinstein consider the case N = 1 and p = 3. For a given nondegenerate critical point of the potential V, assumed globally bounded, and for 0 < E < inf V, they construct a standing wave provided that h is sufficiently small. This solution concentrates around the critical point as h --+ 0. Their method, based on an interesting Lyapunov-Schmidt reduction, was ex- tended by Oh in [6], [7] to conclude a similar result in higher dimensions, pro- N+2 vided that 1 < p < g'=l" He restricts himself to potentials with "mild oscillation" at infinity, namely belonging to a Kato class. In case that V is bounded this re- striction is not necessary as observed by Wang in [10]. * Partially supported by NSF and Grant CI1"CT93-0323 CCE. *'~ Partially supported by FONDECYT Grant No 1212-91 and Fundaci6n Andes.