03-266 Nils Ackermann
On a Periodic Schr\"odinger Equation with Nonlocal Superlinear Part (150K, pdf) Jun 4, 03
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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 $N\ge2$.

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