## On a periodic Schrödinger equation with nonlocal superlinear part.(English)Zbl 1059.35037

Summary: We consider the Choquard-Pekar equation $-\Delta u + Vu = \left(W*u^2\right)u \qquad u \in H^1(\mathbb R^3)$ 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 $$N \geq 2$$.

### MSC:

 35J60 Nonlinear elliptic equations 35Q40 PDEs in connection with quantum mechanics 35J20 Variational methods for second-order elliptic equations 35B10 Periodic solutions to PDEs 49J35 Existence of solutions for minimax problems 81V70 Many-body theory; quantum Hall effect
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