Geodesic
Multiple semiclassical states for singular magnetic nonlinear Schrödinger equations
Electronic journal of differential equations, Tome 2008 (2008)
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},
     year = {2008},
     volume = {2008},
     zbl = {1172.35369},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a38/}
}
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%J Electronic journal of differential equations
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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/