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We establish existence of infinitely many distinct solutions to the semilinear elliptic Hartree-Fock equations for

In the present paper we prove existence of infinitely many solutions to the quasirelativistic Hartree-Fock equations

The novelty of the present paper is Theorem

In the process of implementing these ideas we have to overcome additional technicalities for the quasirelativistic setting compared to the nonrelativistic, for instance, the Coulomb potential is not relatively compact (in the operator sense) with respect to the quasirelativistic energy operator. In particular, compact Sobolev imbeddings are not available (for a recent survey of such problems, we refer to Bartsch et al. [

In the opposite direction, Lieb [

We invoke a direct method developed by Fang and Ghoussoub [

Work related to our study of semilinear elliptic equations and critical point theory includes existence of solutions with finite Morse indices established by Dancer [

Throughout the paper we denote by

For

Let

We need the following abstract operator result by Lions [

Let

Within the Born-Oppenheimer approximation, the quantum energy of

In fact, if

By standard arguments (see, e.g., [

The functional

Let

By

If

We prove the above-mentioned form boundedness. It follows from the following inequality (first observed, it seems, by Kato [

We can re-express

Herein we introduce the quasirelativistic Fock operator.

Assume

Bear in mind the definitions of

We will later need the following spectral result.

Assume

By a minor modification of [

Within the nonrelativistic context a similar result was first given by Lions [

In this section we give the main auxiliary result that will be used in the proof of Theorem

Assume that

there exists a sequence of positive reals

Moreover, the components of the limit element

First we treat the case

Let us now extract some subsequences that we will need. Let us start by proving existence of

Finally, we consider the case

The density operator argument in the proof of Proposition

It is worth to mention that from the perspective of Physics, there is no difference between the requirements

The main result is the following theorem.

Assume that the total nuclear charge

Before proving assertion

First of all we note that using (

Proceeding towards the second assertion of Theorem

We will prove that there exists a critical point at infinitely many distinct levels. We will use abstract critical point theory by Fang and Ghoussoub [

Let us now prove the properties of the sequence

The regularity and decay properties of our sequence were proved in [

The research of the second author is supported by a Stokes Award (Science Foundation Ireland).