September  2010, 9(5): 1263-1281. doi: 10.3934/cpaa.2010.9.1263

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

1. 

Dipartimento di Matematica, Politecnico di Bari, via Orabona 4, 70125 Bari, Italy

2. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510 México D.F., Mexico

Received  August 2009 Revised  November 2009 Published  May 2010

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.

Citation: Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263
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