Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX MULTI-PEAK STANDING WAVES FOR NONLINEAR SCHR ¨ ODINGER EQUATIONS WITH A GENERAL NONLINEARITY Jaeyoung Byeon Department of Mathematics, Pohang University of Science and Technology Pohang, Kyungbuk 790-784, Republic of Korea Louis Jeanjean Equipe de Math´ematiques (UMR CNRS 6623), Universit´e de Franche-Comt´e, 16 Route de Gray, 25030 Besan¸ con, France (Communicated by Aim Sciences) Abstract. We consider singularly perturbed elliptic equations ε 2 Δu-V (x)u+ f (u)=0,x ∈ R N ,N ≥ 3. For small ε> 0, we glue together localized bound state solutions concentrating at isolated components of positive local minimum of V under conditions on f we believe to be almost optimal. 1. Introduction. This paper deals with the study of standing waves for the non- linear Schr¨ odinger equation i ∂ψ ∂t +  2 2 Δψ − V (x)ψ + f (ψ)=0, (t,x) ∈ R × R N . (1) Here  denotes the Plank constant, i the imaginary unit. For the physical back- ground of this equation, we refer to the introduction in [7]. We assume that f (exp(iθ)v) = exp(iθ)f (v) for v ∈ R. A standing wave is a solution of the form ψ(x,t) = exp(−iEt/)v(x). Then, ψ(x,t) is a solution of (1) if and only if the function v satisfies  2 2 Δv − (V (x) − E)v + f (v)=0 in R N . (2) We are interested in positive solutions in H 1 (R N ) for small  > 0. For small  > 0, these standing waves are referred as semi-classical states. For simplicity and without loss of generality, we write V − E as V , i.e., we shift E to 0. Thus, we consider the following equation ε 2 Δv − V (x)v + f (v)=0, v> 0,v ∈ H 1 (R N ) (3) when ε> 0 is sufficiently small. Throughout the paper, the potential V will be assumed to satisfy (V1) V ∈ C(R N , R), 0 ≤ V 0 ≡ inf R N V (x) and lim inf |x|→∞ V (x) > 0. 2000 Mathematics Subject Classification. Primary: 35J20; Secondary: 35J60. Key words and phrases. Variational methods, singularly perturbed elliptic problems. 1