Communications on Pure and Applied Analysis (CPAA)

Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory

Pages: 1263 - 1281, Volume 9, Issue 5, September 2010      doi:10.3934/cpaa.2010.9.1263

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Silvia Cingolani - Dipartimento di Matematica, Politecnico di Bari, via Orabona 4, 70125 Bari, Italy (email)
Mónica Clapp - Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510 México D.F., Mexico (email)

Abstract: We consider the magnetic NLS equation

$ (-\varepsilon i \nabla+A(x)) ^2 u+V(x)u=K(x) |u|^{p-2}u, \quad x\in R^N, $

where $N \geq 3$, $2 < p < 2^*: = 2N/(N-2)$, $A:R^N\to R^N$ is a magnetic potential and $V: R^N \to R$, $K:R^N \to R$ 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 $\tau:G\to S^1$ into the unit complex numbers we look for semiclassical solutions $u_{\varepsilon}: R^N\to C$ to the above equation which satisfy

$ u_{\varepsilon}(gx)=\tau(g)u_{\varepsilon}(x)$

for all $g\in G$, $x\in R^N$. Using equivariant Morse theory we obtain a lower bound for the number of solutions of this type.

Keywords:  Nonlinear Schrödinger equation, magnetic field, equivariant Morse theory, symmetric semiclassical states.
Mathematics Subject Classification:  35J20, 35Q40, 35Q55, 37K40, 83C40.

Received: August 2009;      Revised: November 2009;      Available Online: May 2010.