Nils Ackermann
On a Periodic Schr\"odinger Equation with Nonlocal Superlinear Part
(150K, pdf)
ABSTRACT. We consider the Choquard-Pekar equation
\begin{equation*}
-\Delta u +Vu=(W*u^2)u\qquad u\in H^1(\dR^3)
\end{equation*}
and focus on the case of periodic potential $V$. For a large class
of even functions $W$ we show existence and multiplicity of
solutions. Essentially the conditions are that $0$ is not in the
spectrum of the linear part $-\Delta+V$ and that $W$ does not change
sign. Our results carry over to more general nonlinear terms in
arbitrary space dimension $N\ge2$.