Existence Results for the Periodic Thomas-Fermi-Dirac-von Weizsäcker Equations

Shaowei Chen, Lishan Lin, and Liqin Xiao 1School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China 2School of Applied Mathematical Sciences, Xiamen University of Technology, Xiamen 361024, China Correspondence should be addressed to Shaowei Chen; chensw@amss.ac.cn Received 18 October 2014; Accepted 23 December 2014 Academic Editor: Hagen Neidhardt Copyright © 2015 Shaowei Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction and Statement of Results
In this paper, we consider the following equation: where 2 <  <  < 6,  > 0 is a parameter, and  1 (R 3 ) is the standard Sobolev space with norm The potential function  ∈  ∞ (R  ) is 1-periodic in   for  = 1, 2, 3.Under this assumption, (), the spectrum of the operator is a purely continuous spectrum that is bounded below and consists of closed disjoint intervals ([1, Theorem XIII.100]).Thus, the complement R \ () consists of open intervals called spectral gaps.We assume the following: (v)  ∈  ∞ (R  ) is 1-periodic in   for  = 1, 2, 3 and 0 is in a spectral gap (−, ) of −Δ + , where 0 < ,  < +∞.
A solution  of (1) is called nontrivial if  ̸ ≡ 0. Our main result is as follows.
Equation (1) arises in the study of the Thomas-Fermi-Dirac-von Weizsäcker (TFDW) model for atoms and molecules with no external potential, where  =  2 is the electron density.In [2, section VIII], Lieb studied the existence and symmetric and asymptotic properties of solutions to − Δ + ( + )  + ( 2 ⋆ || −1 )  +  2−1 −    5/3   = − in R 3 (4) for various choices of .Here, , , ,   , and  are constants and  > 4/3.The term −   5/3 is called the Dirac term 2 Advances in Mathematical Physics with strength   .In [3], Le Bris considered the minimizing problem inf with where  1 and  2 are positive constant, and  is the Coulomb potential created by the atomic nuclei.Le Bris proved that there exists a   > 0, such that, for 0 <  <   , the problem admits a minimizer.Moreover, if  2 is small enough, then the minimizer is unique.Using the Lagrange multiplier, it is easy to see that the minimizer of ( 5) is a solution to for some .However, in a recent paper [4] by Lu and Otto, the authors proved that for sufficiently large , the variational problem (5) with  = 0 does not have a minimizer.Equation (1) with  a periodic function is used to describe a Hartree model for crystals (see [5,6]).Moreover, when  = 0, (1) is often referred to the Thomas-Fermi-von Weizsäcker model in the literature.One can see [7] and references therein.Finally, we should mention that the so-called Schrödinger-Poisson-Slater equation is also related to (1) and has attracted much attention in recent years (see [5,[7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]).
The variational functional for (1) is given by In other words, the critical points of   are solutions to (1).Hence, it is natural to use critical point theory to obtain solutions to (1).Under assumption (v), the quadratic form ∫ R 3 (|∇| 2 +  2 ) has infinite-dimensional positive and negative spaces.It can be shown that for sufficiently small  > 0, −  has a global infinite-dimensional linking geometry (for its definition, see (6.4) of Willem [23]).However, the difficulty rises when the classical infinite-dimensional linking theorem (see [23]) is used to obtain a critical point to   .This theorem requires the functional to satisfy some upper semicontinuous assumption (see (6.3) of [23]).However, because the nonlinearity is neither positive definite nor negative definite in  1 (R 3 ) whenever  > 0, −  does not satisfy the upper semicontinuous assumption.To overcome this difficulty, we use a new infinite-dimensional linking theorem in [24] to obtain a ()  sequence (see Definition 6) for −  .We can prove that a ()  sequence for −  is bounded in  1 (R 3 ) if  > 0 is sufficiently small.This result is new and original.Finally, through the concentration-compactness principle and the ()  sequence, a nontrivial solution to (1) is obtained.Our method can be used to study more general equation like Notation.  () denotes an open ball of radius  and center .For a Banach space , we denote the dual space of  by   and denote strong and weak convergence in  by → and ⇀, respectively.For  ∈  1 (; R), we denote the Fréchet derivative of  at  by   ().The Gateaux derivative of  is denoted by ⟨  (), V⟩, for all , V ∈ .  (R 3 ) and   loc (R 3 ) denote the standard   space and the locally -integrable function space, respectively (1 ≤  ≤ ∞).Let Ω be a domain in R  ( ≥ 1). ∞ 0 (Ω) is the space of infinitely differentiable functions with compact support in Ω.We use (ℎ) and (ℎ) to mean |(ℎ)| ≤ |ℎ| and (ℎ)/|ℎ| → 0, respectively.

Variational Setting for (1)
Let D 1,2 (R 3 ) be the Hilbert space with inner product For  ∈  1 (R 3 ), by the Lax-Milgram theorem, the equation has a unique solution   ∈ D 1,2 (R 3 ) (see Proposition 2.2 of [10]).And, by Theorem 2.2.1 of [25],   can be expressed by The function   has the following properties.
By this lemma, the functional   in ( 9) is well defined in  1 (R 3 ).And a direct computation shows that the derivative of   is It is easy to verify that   is a  1 functional in  1 (R 3 ).Moreover, we have the following.
Lemma 3. The following statements are equivalent: ) is a solution of ( 1); Under the assumption (k), there is a standard variational setting for the quadratic form ∫ R 3 (|∇| 2 +() 2 ).One can see section 6.4 of [23].But for the convenience of the reader, we state it here.
Let  be the operator defined by (3).We denote by || 1/2 the square root of the absolute value of .The domain of || 1/2 is the space On , we choose the inner product V  and the corresponding norm ‖‖ = √(, ).Since 0 lies in a gap of the essential spectrum of , there exists an orthogonal decomposition  =  ⊕  such that  and  are the positive and negative spaces corresponding to the spectral decomposition of .Since  is 1-periodic for all variables, they are invariant under the action of Z 3 , that is, for any  ∈  or  ∈  and for any k Let  :  → ,  :  →  be the orthogonal projections.By (21), Moreover, by  =  ⊕ , we have By ( 22) and ( 9), Moreover, by (19),

Boundedness of (𝐶)
In this section, we will prove that if  > 0 is sufficiently small, then a ()  sequence for −  is bounded in .

Proof of Theorem 1
Proof of Theorem    , we have that −   ( 0 ) = 0. Therefore,  0 is a nontrivial solution of (1).This completes the proof.