| Abstract: |
| The Schr\odinger-Poisson equation has been first introduced in dimension $N=3$ in 1954
by Pekar to describe quantum theory of a polaron at rest, and then applied by Choquard in 1976 as an approximation
to the Hartree-Fock theory of one-component plasma.
It has been extensively studied in the higher dimensional case $N \geq 3$, due to the richness of plenty of
applications and to the new mathematical challenges related to nonlocal problems.
On the other hand, the literature is not abundant for the planar case $N=2$, due to the presence of a sign-changing and unbounded
logarithmic integral kernel, which demands for new functional settings where implementing the variational approach.\
We review here some recent results on this topic and on some new related inequalities. |
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