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Hartree-Fock and related equations. (English) Zbl 0693.35047
Nonlinear partial differential equations and their applications, Lect. Coll. de France Semin., Vol. IX, Paris/Fr. 1985-86, Pitman Res. Notes Math. Ser. 181, 304-333 (1988).
[For the entire collection see Zbl 0653.00012.]
The author presents some results concerning several equations of mathematical physics of Hartree-Fock-type. The common feature is to study critical points of the functional $E(\phi)=\int_{{\mathbb{R}}^ 3}(| \nabla \phi (x)|^ 2+F(| \phi (x)|^ 2)+V_ 0(x)| \phi (x)|^ 2)dx$ $$+\frac{1}{4}\iint_{{\mathbb{R}}^ 3\times {\mathbb{R}}^ 3}(| \phi (x)|^ 2| \phi (y)|^ 2-(| <\phi (x),\phi (y)>|^ 2\cdot V(x-y)dx dy$$on the set $$K=\{\phi \in H^ 1_ 2({\mathbb{R}}^ 3)^ N|\int_{{\mathbb{R}}^ 3}\phi_ i\overline{\phi_ j}dx=\delta_{ij}\}$$, where F, $$V_ 0$$ and V are given. Typically $$V(x)=| x|^{-1}$$ and $$V_ 0(x)=-\sum^{m}_{j=1}z_ j| x-x_ j|^{-1}$$ in atomic physics or $$V_ 0=0$$ and $$V\in L_ 1\cap L_ p$$ in nuclear physics. Though one lacks the Palais-Smale condition, one may find critical points $$u^ R$$ of approximating problems on $$K_ R=K\cap \overset\circ H^ 1_ 2(B_ R)^ N$$ with uniformly bounded Morse index, which in turn allows to conclude compactness in $$H^ 1_ 2$$. For further results and more details see the author’s paper [Commun. Math. Phys. 109, 33-97 (1987; Zbl 0618.35111)].
Reviewer: M.Wiegner

##### MSC:
 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 35Q99 Partial differential equations of mathematical physics and other areas of application 35D05 Existence of generalized solutions of PDE (MSC2000)