Advances in Differential Equations Volume 11, Number 7 (2006), 813–840 AN EXISTENCE AND STABILITY RESULT FOR STANDING WAVES OF NONLINEAR SCHR ¨ ODINGER EQUATIONS Louis Jeanjean and Stefan Le Coz Equipe de Math´ ematiques (UMR CNRS 6623) Universit´ e de Franche-Comt´ e 16 Route de Gray, 25030 Besan¸ con, France (Submitted by: Reza Aftabizadeh) Abstract. We consider a nonlinear Schr¨odinger equation with a non- linearity of the form V (x)g(u). Assuming that V (x) behaves like |x| −b at infinity and g(s) like |s| p around 0, we prove the existence and orbital stability of travelling waves if 1 <p< 1 + (4 − 2b)/N . 1. Introduction This paper concerns the existence and orbital stability of standing waves for the nonlinear Schr¨ odinger equation iu t +Δu = V (x)g(u), (t, x) ∈ R × R N ,N  3. (1.1) Here u ∈ H 1 (R N , C), V is a real-valued potential and g is a nonlinearity satisfying g(e iθ s)= e iθ g(s) for s ∈ R. A solution of the form u(t, x)= e iλt ϕ(x), where λ ∈ R, is called a standing wave. For solutions of this type with ϕ ∈ H 1 (R N , R), (1.1) is equivalent to −Δϕ + λϕ = V (x)g(ϕ), ϕ ∈ H 1 (R N , R). (1.2) We are interested in the existence of positive solutions of (1.2) for small λ> 0. In addition we study the stability of the corresponding solutions of (1.1). In the autonomous case, i.e., when V is a constant, we refer to the fun- damental papers of Berestycki and Lions [2] where sufficient and almost necessary conditions are derived for the existence in H 1 (R N , R) of a solution of (1.2). When (1.2) is nonautonomous only partial results are known. A major difficulty to overcome is the lack of a priori bounds for the solutions. In contrast to the autonomous case, where using dilations and taking ad- vantage of the Pohozaev identity is at the heart of the results of [2], no such Accepted for publication: April 2006. AMS Subject Classifications: 35J60, 35Q55, 37K45, 35B32. 813