Electron. J. Diff. Eqns., Vol. 2008(2008), No. 37, pp. 1-18.

Multiple semiclassical states for singular magnetic nonlinear Schrodinger equations

Sara Barile

Abstract:
By means of a finite-dimensional reduction, we show a multiplicity result of semiclassical solutions $u: \mathbb{R}^N \to\mathbb{C}$ to the singular nonlinear Schrodinger equation
$$ \Big( \frac{\varepsilon}{i} \nabla - A(x)\Big)^2 u + u+(V(x)-\gamma(\varepsilon)W(x)) u = K(x) | u|^{p-1} u, \quad x \in \mathbb{R}^N, $$
where $N \geq 2$, $1 < p < 2^{*}-1$, $A(x), V(x)$ and $K(x)$ are bounded potentials. Such solutions concentrate near (non-degenerate) local extrema or a (non-degenerate) manifold of stationary points of an auxiliary function $\Lambda$ related to the unperturbed electric field $V(x)$ and the coefficient $K(x)$ of the nonlinear term.

Submitted November 26, 2007. Published March 14, 2008.
Math Subject Classifications: 35J10, 35J60, 35J20, 35Q55, 58E05.
Key Words: Nonlinear Schrodinger equations; external magnetic field; singular potentials; semiclassical limit.

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Sara Barile
Dipartimento di Matematica, Politecnico di Bari
Via Orabona 4, I-70125 Bari, Italy
email: s.barile@dm.uniba.it

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