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, Silvia;
2010

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.
6th European Conference on Elliptic and Parabolic Problems
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11589/18599
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