Spring School at AIMS Senegal , M ’ bour , February 15 - 20 , 2015 on Variational and Geometric methods in Nonlinear PDEs

Goals of this lecture-series: • Discuss existence of standing waves for the nonlinear Schrödinger equation (NLS) (1) i ∂w ∂t = −∆w + ˜ V (x)w − Γ(x)|w| p−1 w, x ∈ R n , t ∈ R for functions w : R n × R → C and for p > 1. • Discuss aspects of symmetry breaking and bifurcation. 1. Standing waves and their variational formulation A solution w(x, t) = u(x)e −iωt with u decaying to 0 at infinity is called standing wave. It has to satisfy (2) −∆u + (˜ V (x) − ω) =:V (x) u = Γ(x)|u| p−1 u, x ∈ R n with u(x) → 0 as |x| → ∞.