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

Ackermann, Nils

Math. Z. 248, No. 2, 423-443 (2004).

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

Keywords:

Choquard-Pekar equation; periodic potential; existence; multiplicity of solutions; critical points; energy functional; Hartree equation