Geodesic
Multiple semiclassical states for singular magnetic nonlinear Schrödinger equations
Electronic Journal of Differential Equations, Tome 2008 (2008).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: By means of a finite-dimensional reduction, we show a multiplicity result of semiclassical solutions $$ \Big( \frac{\varepsilon}{i} \nabla - A(x)\Big)^2 u + u+(V(x)-\gamma(\varepsilon)W(x)) u = K(x) | u|^{p-1} u, \quad x \in \mathbb{R}^N, $$ 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.
Classification : 35J10, 35J60, 35J20, 35Q55, 58E05
Keywords: nonlinear Schrödinger equations, external magnetic field, singular potentials, semiclassical limit 
@article{EJDE_2008__2008__a38,
     author = {Barile, Sara},
     title = {Multiple semiclassical states for singular magnetic nonlinear {Schr\"odinger} equations},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2008},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a38/}
}
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Barile, Sara. Multiple semiclassical states for singular magnetic nonlinear Schrödinger equations. Electronic Journal of Differential Equations, Tome 2008 (2008). http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a38/