SYMMETRIC SEMICLASSICAL STATES TO A MAGNETIC NONLINEAR SCHR ¨ ODINGER EQUATION VIA EQUIVARIANT MORSE THEORY SILVIA CINGOLANI AND M ´ ONICA CLAPP Abstract. We consider the magnetic NLS equation (−εi∇ + A(x)) 2 u + V (x)u = K(x) |u| p-2 u, x ∈ R N , where N ≥ 3, 2 <p< 2 * := 2N/(N − 2), A : R N → R N is a magnetic potential and V : R N → R, K : R N → R are bounded positive potentials. We consider a group G of orthogonal trans- formations of R N and we assume that A is G-equivariant and V , K are G-invariant. Given a group homomorphism τ : G → S 1 into the unit complex numbers we look for semiclassical solutions u ε : R N → C to the above equation which satisfy u ε (gx)= τ (g)u ε (x) for all g ∈ G, x ∈ R N . Using equivariant Morse theory we obtain a lower bound for the number of solutions of this type. 1. Introduction Let G be a closed subgroup of the group O(N ) of linear isometries of R N . We assume A : R N → R N is a C 1 -function and V,K : R N → R are bounded C 2 -functions with inf R N V> 0 and inf R N K> 0 which satisfy A(gx)= gA(x) ∀g ∈ G, x ∈ R N , (1.1) V (gx)= V (x),K (gx)= K (x) ∀g ∈ G, x ∈ R N . Given a group homomorphism τ : G → S 1 into the unit complex num- bers we look for solutions u : R N → C to the problem (℘ ε )    (−εi∇ + A) 2 u + Vu = K |u| p−2 u, u ∈ L 2 (R N , C), ε∇u + iAu ∈ L 2 (R N , C N ), S. Cingolani is supported by the MIUR project “Variational and topological methods in the study of nonlinear phenomena” (PRIN 2008). M. Clapp is supported by CONACYT grant 58049 and PAPIIT grant IN101209. 1