SYMMETRIC SEMICLASSICAL STATES TO A MAGNETIC NONLINEAR SCHRÖDINGER EQUATION VIA EQUIVARIANT MORSE

Abstract. We consider the magnetic NLS equation (−εi∇+A(x))2 u+ V (x)u = K(x) |u|p−2 u, x ∈ RN, where N ≥ 3, 2 < p < 2 ∗: = 2N/(N − 2), A: RN → RN is a magnetic potential and V: RN → R, K: RN → R are bounded positive potentials. We consider a group G of orthogonal trans-formations of RN and we assume that A is G-equivariant and V, K are G-invariant. Given a group homomorphism τ: G → S1 into the unit complex numbers we look for semiclassical solutions uε: RN → C to the above equation which satisfy uε(gx) = τ(g)uε(x) for all g ∈ G, x ∈ RN. Using equivariant Morse theory we obtain a lower bound for the number of solutions of this type. 1