Symmetric Semiclassical States to a Magnetic Nonlinear Schrödinger Equation via Equivariant Morse Theory

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