SYMMETRIC SEMICLASSICAL STATES TO A MAGNETIC NONLINEAR SCHRODINGER EQUATION VIA EQUIVARIANT MORSE THEORY

We consider the magnetic NLS equation (-epsilon i del + A(x))(2) u + V(x)u = K(x)|u|(p-2)u, x is an element of 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 transformations of R-N and we assume that A is G-equivariant and V, K are G-invariant. Given a group homomorphism tau : G --> S-1 into the unit complex numbers we look for semiclassical solutions u(epsilon): R-N --> C to the above equation which satisfy u(epsilon)(gx) = tau(g)u(epsilon)(x) for all g is an element of G, x is an element of R-N. Using equivariant Morse theory we obtain a lower bound for the number of solutions of this type.