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 $N\ge2$.