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Mathematics  2009 

A note on Schr?dinger--Newton systems with decaying electric potential

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Abstract:

We prove the existence of solutions for the singularly perturbed Schr\"odinger--Newton system {ll} \hbar^2 \Delta \psi - V(x) \psi + U \psi =0 \hbar^2 \Delta U + 4\pi \gamma |\psi|^2 =0 . \hbox{in $\mathbb{R}^3$} with an electric potential (V) that decays polynomially fast at infinity. The solution $\psi$ concentrates, as $\hbar \to 0$, around (structurally stable) critical points of the electric potential. As a particular case, isolated strict extrema of (V) are allowed.

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