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 transformations 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.
Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory
Cingolani S
;
2010-01-01
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 transformations 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
