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 \(N \geq 2\).


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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