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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)