An eigenvalue semiclassical problem for the Schrödinger operator with an electrostatic field

Teresa D'Aprile

DOI: http://dx.doi.org/10.12775/TMNA.2006.006

Abstract


We consider the following system of Schrödinger-Maxwell
equations in the unit ball $B_1$ of ${\mathbb R}^3$
$$
-\frac{\hbar^2}{2m}\Delta v+ e\phi v=\omega v,
\quad
-\Delta\phi=4\pi e v^2
$$
with the boundary conditions $ u=0$, $
\phi=g$ on $\partial B_1$, where $\hbar$, $m$, $e$,
$\omega > 0$, $v$, $\phi\colon B_1\rightarrow {\mathbb R}$, $g\colon \partial B_1\to {\mathbb R}$.
Such system describes the interaction of a particle constrained to
move in $B_1$ with its own electrostatic field. We exhibit a family
of positive solutions $(v_\hbar, \phi_\hbar)$ corresponding
to eigenvalues $\omega_\hbar$ such that $v_\hbar$ concentrates
around some points of the boundary $\partial B_1$ which are
minima for $g$ when $\hbar\rightarrow 0$.

Keywords


Schrödinger-Maxwell system; existence; concentration

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