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

Authors: Shaowei Chen; Lishan Lin; Liqin Xiao
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

We consider the Thomas-Fermi-Dirac-von Weizsäcker equation-Δu+Vxu+u2⋆(1/x)u = λup-2u-uq-2u, u∈H1R3, where λ>0 is a parameter, 2<p<q<6, V∈L∞(R3) is 1-periodic in xj, for j=1,2,3, and 0 is in a spectral gap of the operator -Δ+V. Using a new infinite-dimensional linking theorem, we prove that, for sufficiently small λ>0, this equation has a nontrivial solution.


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## Body

## 1. Introduction and Statement of Results

In this paper, we consider the following equation:(1)-Δu+Vxu+u2⋆1xu=λup-2u-uq-2u,u∈H1R3,
where 2<p<q<6,  λ>0 is a parameter,
(2)u2⋆1x=∫R3‍u2yx-ydy,
and H1(R3) is the standard Sobolev space with norm uH1(R3)=(∫R3‍(|∇u|2+u2)dx)1/2.The potential functionV∈L∞(RN) is 1-periodic in xj for j=1,2,3. Under this assumption, σ(L), the spectrum of the operator
(3)L=-Δ+V:L2R3⟶L2R3
is a purely continuous spectrum that is bounded below and consists of closed disjoint intervals ([1, Theorem  XIII.100]). Thus, the complement R∖σ(L) consists of open intervals called spectral gaps. We assume the following:(v)
V∈L∞(RN) is 1-periodic in xj for j=1,2,3 and 0 is in a spectral gap (-α,β) of -Δ+V, where 0<α,β<+∞.A solutionu of (1) is called nontrivial if u≢0. Our main result is as follows.Theorem 1.
Suppose that2<p<q<6 and (v) is satisfied. Then, there exists λ0>0 such that, for any 0<λ<λ0, the problem (1) has a nontrivial solution.Equation (1) arises in the study of the Thomas-Fermi-Dirac-von Weizsäcker (TFDW) model for atoms and molecules with no external potential, where ρ=u2 is the electron density. In [2, section VIII], Lieb studied the existence and symmetric and asymptotic properties of solutions to
(4)-AΔψ+V+αψ+ψ2⋆x-1ψ+γψ2p-1-Ceψ5/3=-μψinR3
for various choices of V. Here, A,α,γ,Ce, and μ are constants and p>4/3. The term -Ceψ5/3 is called the Dirac term with strength Ce. In [3], Le Bris considered the minimizing problem
(5)inf⁡ψ∈H1(R3),∫R3‍ψ2dx=λEψ
with
(6)Eψ=∫R3‍∇ψ2dx+∫R3‍Vxψ2dx+c1∫R3‍ψ10/3dx-c2∫R3‍ψ8/3dx+∫R3‍∫R3‍ψ2xψ2yx-ydx,
where c1 and c2 are positive constant, and V is the Coulomb potential created by the atomic nuclei. Le Bris proved that there exists a λc>0, such that, for 0<λ<λc, the problem admits a minimizer. Moreover, if c2 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
(7)-Δψ+Vxψ+ψ2⋆x-1ψ+c1ψ7/3-c2ψ5/3=μψinR3
for some μ. However, in a recent paper [4] by Lu and Otto, the authors proved that for sufficiently large λ, the variational problem (5) with V=0 does not have a minimizer. Equation (1) with V a periodic function is used to describe a Hartree model for crystals (see [5, 6]). Moreover, when λ=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
(8)-Δu+Vxu+u2⋆1xu=λup-2u,inR3
is also related to (1) and has attracted much attention in recent years (see [5, 7–22]).The variational functional for (1) is given by
(9)Jλu=12∫R3‍∇u2+Vxu2dx+14∫R3‍∫R3‍u2xu2yx-ydxdy-λp∫R3‍updx+1q∫R3‍uqdx,u∈H1R3.
In other words, the critical points of Jλ are solutions to (1). Hence, it is natural to use critical point theory to obtain solutions to (1). Under assumption (v), the quadratic form ∫R3‍(|∇u|2+Vu2)dx has infinite-dimensional positive and negative spaces. It can be shown that for sufficiently small λ>0, -Jλ 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 Jλ. This theorem requires the functional to satisfy some upper semicontinuous assumption (see (6.3) of [23]). However, because the nonlinearity
(10)14∫R3‍∫R3u2xu2yx-ydxdy+1q∫R3‍uqdx-λp∫R3‍updx
is neither positive definite nor negative definite in H1(R3) whenever λ>0, -Jλ does not satisfy the upper semicontinuous assumption. To overcome this difficulty, we use a new infinite-dimensional linking theorem in [24] to obtain a (C¯)c sequence (see Definition 6) for -Jλ. We can prove that a (C¯)c sequence for -Jλ is bounded in H1(R3) if λ>0 is sufficiently small. This result is new and original. Finally, through the concentration-compactness principle and the (C¯)c sequence, a nontrivial solution to (1) is obtained. Our method can be used to study more general equation like
(11)-Δu+Vxu+u2⋆1xu=fx,u,u∈H1R3.Notation. Br(a) denotes an open ball of radius r and center a. For a Banach space X, we denote the dual space of X by X′ and denote strong and weak convergence in X by → and ⇀, respectively. For φ∈C1(X;R), we denote the Fréchet derivative of φ at u by φ′(u). The Gateaux derivative of φ is denoted by φ′u,v, for all u,v∈X.  Lq(R3) and Lloc⁡q(R3) denote the standard Lq space and the locally q-integrable function space, respectively (1≤q≤∞). Let Ω be a domain in RN (N≥1). C0∞(Ω) is the space of infinitely differentiable functions with compact support in Ω. We use O(h) and o(h) to mean |O(h)|≤C|h| and o(h)/|h|→0, respectively.


## 2. Variational Setting for (1)

LetD1,2(R3) be the Hilbert space
(12)D1,2R3=u∈L6R3∣∫R3‍∇u2dx<∞
with inner product
(13)u,vD1,2(R3)=∫R3‍∇u∇vdx.
For u∈H1(R3), by the Lax-Milgram theorem, the equation
(14)-Δϕ=4πu2inR3
has a unique solution ϕu∈D1,2(R3) (see Proposition  2.2 of [10]). And, by Theorem  2.2.1 of [25], ϕu can be expressed by
(15)ϕux=u2⋆1y(x)=∫R3‍u2yx-ydy,x∈R3.
The function ϕu has the following properties.Lemma 2.
(i) There exists a positive constantC such that, for any u∈H1(R3),
(16)ϕuD1,2(R3)≤CuH1R32,(17)∫R3‍∇ϕu2dx=4π∫R3‍ϕuu2dx=4π∫R3‍∫R3‍u2xu2yx-ydxdy≤CuL12/5(R3)4≤CuH1(R3)4.(ii) For anyu∈H1(R3), ϕu≥0 in R3.Proof.
The inequality(18)4π∫R3‍∫R3‍u2xu2yx-ydxdy≤CuL12/5(R3)4
can be derived from the the Hardy-Littlewood-Sobolev inequality (see (1.1) of [26]) by choosing n=3,  λ=1,  p=t=6/5, and f=g=u2 there. For a proof of the other properties of ϕu in this lemma, one can see [8, 10, 12].By this lemma, the functionalJλ in (9) is well defined in H1(R3). And a direct computation shows that the derivative of Jλ is
(19)Jλ′u,v=∫R3‍∇u∇v+Vxuvdx+∫R3‍∫R3‍uxvxu2yx-ydxdy-λ∫R3‍up-2uvdx+∫R3‍uq-2uvdx,xxxxxxxxxxxxxxxxx∀u,v∈H1R3.
It is easy to verify that Jλ is a C1 functional in H1(R3). Moreover, we have the following.Lemma 3.
The following statements are equivalent:(i)
u∈H1(R3) is a solution of (1);(ii)
u is a critical point of Jλ.Under the assumption(v), there is a standard variational setting for the quadratic form ∫R3‍(|∇u|2+V(x)u2)dx. One can see section 6.4 of [23]. But for the convenience of the reader, we state it here.LetL be the operator defined by (3). We denote by |L|1/2 the square root of the absolute value of L. The domain of |L|1/2 is the space
(20)X:=H1R3.
On X, we choose the inner product (u,v)=∫R3‍|L|1/2u·|L|1/2vdx and the corresponding norm u=(u,u). Since 0 lies in a gap of the essential spectrum of L, there exists an orthogonal decomposition X=Y⊕Z such that Y and Z are the positive and negative spaces corresponding to the spectral decomposition of L. Since V is 1-periodic for all variables, they are invariant under the action of Z3, that is, for any u∈Y or u∈Z and for any k=(n1,n2,n3)∈Z3, u(·-k) is also in Y or Z. Furthermore,
(21)∀u∈Y,∫R3‍∇u2+Vu2dx=u,u=u2,∀u∈Z,∫R3‍∇u2+Vu2dx=-u,u=-u2.
Let Q:X→Z, P:X→Y be the orthogonal projections. By (21),
(22)∫R3‍∇u2+Vu2dx=Pu2-Qu2,∀u∈X.
Moreover, by X=Y⊕Z, we have
(23)u=Pu+Qu,u2=Pu2+Qu2,∀u∈X.By (22) and (9),
(24)-Jλu=12Qu2-12Pu2-14∫R3∫R3u2xu2yx-ydxdy+λp∫R3‍updx-1q∫R3‍uqdx,u∈X.
Moreover, by (19),
(25)-Jλ′u,v=Qu,v-Pu,v-∫R3‍∫R3‍uxvxu2yx-ydxdy+λ∫R3‍up-2uvdx-∫R3‍uq-2uvdx,∀u,v∈X.


## 3. A Global Linking Geometry for-Jλ

Let{ek} be a total orthonormal sequence in Y and
(26)u=max⁡⁡Qu,∑k=1∞‍12k+1Pu,ek.
For R>r>0 and u0∈Z with u0=1, set
(27)N=u∈Z∣u=r,M=u+tu0∣u∈Y,t≥0,u+tu0≤R,∂M=u∈Y∣u≤R∂M=∪u+tu0∣u∈Y,t>0,u+tu0=R.Definition 4.
Letϕ∈C1(X,R). ϕ′ is called weakly sequentially continuous if u∈X and {un}⊂X are such that un⇀u; then, for any φ∈X,  〈ϕ′(un),φ〉→〈ϕ′(u),φ〉.Lemma 5.
The functional-Jλ satisfies the following.(a)
-Jλ′ is weakly sequentially continuous and, for every λ>0,
(28)sup⁡M-Jλ<+∞.(b)
There existδ>0,  R>r>0, u0∈Z with u0=1 and λ0′>0 such that, for 0<λ<λ0′,
(29)inf⁡N-Jλ>max⁡sup⁡∂M-Jλ,sup⁡u≤δ-Jλu.Proof.
(a) Letu∈X and {un}⊂X be such that un⇀u as n→∞. It follows that
(30)Qun,v⟶Qu,v,Pun,v⟶Pu,v,n⟶∞,∀v∈X.
Sinceun⇀u implies un→u in Lloc⁡s(R3) for any 1≤s<6, we get that, for any v∈C0∞(R3), as n→∞,
(31)∫R3‍unp-2unvdx⟶∫R3‍up-2uvdx,∫R3‍unq-2unvdx⟶∫R3‍uq-2uvdx.
By Lemma 2, we have that {ϕun} is bounded in D1,2(R3). Therefore, up to a subsequence, ϕun⇀ϕu as n→∞. It follows that ϕun→ϕu in Lloc⁡s(R3) for any 1≤s<6. This yields that, for any v∈C0∞(R3), as n→∞,
(32)∫R3‍ϕununvdx⟶∫R3‍ϕuuvdx.
From (25), (30), (31), and (32), we deduce that, as n→∞,
(33)-Jλ′un,v⟶-Jλ′u,v,∀v∈C0∞R3.
Hence, -Jλ′ is weakly sequentially continuous. Moreover, it is easy to see that -Jλ maps bounded sets into bounded sets; hence sup⁡M(-Jλ)<+∞.(b)
Ifu∈Z, then Pu=0 and Qu=u. Using (17), we get from (24) that, for any u∈Z,
(34)-Jλu≥12u2-14∫R3‍∫R3‍u2xu2yx-ydxdy-1q∫R3‍uqdx≥12u2-C16πu4-C1quq,where C1 comes from the Sobolev inequality uLq(R3)≤C11/qu, for all u∈X. It follows that we can choose sufficiently small r>0 such that, for N=u∈Z∣u=r,
(35)inf⁡N-Jλ>14r2>0.
Sinceq>p, there exists Λ>0 such that
(36)λptp-1qtq<0if0<λ≤1,t>Λ.
Let C′>0 be such that
(37)uL2(R3)≤C′u,∀u∈X.
From q>p>2, we deduce that there exists 0<λ0′<1 such that, for any 0<λ≤λ0′,
(38)sup⁡t≤Λt-2λptp-1qtq<14C′2.
From (36) and (38), we deduce that, for any 0<λ≤λ0′ and u∈X,
(39)-Jλu=12Qu2-12Pu2-14∫R3‍∫R3‍u2xu2yx-ydxdy+∫R3‍λpup-1quqdx≤12Qu2-12Pu2-14∫R3‍∫R3‍u2xu2yx-ydxdy+∫{x∣ux≤Λ}‍λpup-1quqdx≤12Qu2-12Pu2-14∫R3‍∫R3‍u2xu2yx-ydxdy+14C′2∫R3‍u2dx≤12Qu2-12Pu2-14∫R3‍∫R3‍u2xu2yx-ydxdy+14u2=34Qu2-14Pu2-14∫R3‍∫R3‍u2xu2yx-ydxdy.
From Theorem  1.1 of [27], we deduce that there exists C*>0 such that, for any u∈X,
(40)∫R3‍∫R3‍u2xu2yx-ydxdy≥C*∫R3‍u2x1/21+ln⁡xdx2.
Letu0∈Z be such that u0=1. And let u=v+tu0∈Y⊕Ru0. Since there exists a continuous projection
(41)L2R3,x-1/21+ln⁡x-1⟶Ru0,
we deduce that there exists a constant C**>0 such that, for any u=v+tu0∈Y⊕Ru0,
(42)∫R3‍u2x1/21+ln⁡xdx2≥C**t4.
Combining (39)–(42), we deduce that there exists a constant κ>0 such that, for any u=tu0+v∈Y⊕Ru0,
(43)-Jλu≤34t2-14v2-κt4.
It follows that
(44)-Jλu⟶-∞,asu⟶∞,u∈Y⊕Ru0.
Moreover, for 0<λ≤λ0′ and u∈Y, (39) implies
(45)-Jλ(u)≤-14u2-14∫R3‍∫R3‍u2xu2yx-ydxdy≤0.
Together with (44), this implies that there exists R>r such that
(46)sup⁡∂M-Jλ≤0<inf⁡N-Jλ.
From (39) and the definition of · (see (26)), we have, for 0<λ≤λ0′,
(47)-Jλu≤34Qu2≤34u2.
Choosing δ=r/6, (47) and (35) give that
(48)sup⁡u≤δ-Jλu<inf⁡N-Jλ.
Together with (46), this yields (29).


## 4. Boundedness of(C¯)c Sequence of -Jλ

Definition 6.
Letϕ∈C1(X,R). A sequence {un}⊂X is called a (C¯)c sequence for ϕ, if
(49)sup⁡nϕun≤c,1+unϕ′unX′⟶0,asn⟶∞.In this section, we will prove that ifλ>0 is sufficiently small, then a (C¯)c sequence for -Jλ is bounded in X.Sinceq>p>2, there exists D>0 such that
(50)tq-2-λtp-2>V-L∞R3if0<λ≤1,t>D2,
where V-(x)=max⁡{-V(x),0}, x∈R3.Lemma 7.
Suppose that0<λ≤1. If {un} is a (C¯)c sequence for -Jλ, then
(51)lim⁡n→∞∫{x∣|un(x)|>D}‍un6dx=0.Proof.
Letvn=max⁡{un(x)-D/2,0}. It is easy to verify that vn∈X and vn≤un, for all n∈N. Together with the fact that {un} is a (C¯)c sequence of -Jλ, this implies 〈J′(un),vn〉=o(1), where o(1) denotes the infinitesimal depending only on n; that is, o(1)→0 as n→∞. Choosing v=vn in (19), by (50) and the fact that ϕun,  un, and vn are nonnegative in {x∣un(x)>D/2}, we have
(52)o1=J′un,vn=∫R3‍∇vn2dx+∫R3‍Vxunvndx+∫R3‍ϕun·un·vndx+∫R3‍unq-2-λunp-2unvndx=∫R3‍∇vn2dx+∫R3‍V+unvndx+∫{x∣un(x)>D/2}‍ϕun·un·vndx+∫{x∣unx>D/2}unq-2-λunp-2xxxxxxxxxxxxxxx-V-xunq-2unvndx≥∫R3‍∇vn2dx≥C~∫R3‍vn6dx1/3=C~∫{x∣unx>D/2}‍vn6dx1/3,
where V+=V+V-≥0 in R3 and C~ is the Sobolev constant. Note that vn≥un/2 on {x∣un(x)>D}. Together with (52), this implies
(53)lim⁡n→∞∫{x∣un(x)>D}‍un6dx=0.
Similarly, we can prove
(54)lim⁡n→∞∫{x∣-un(x)>D}‍un6dx=0.
The result of this lemma follows from these two limits.Lemma 8.
There existsλ0′′>0 such that if 0<λ<λ0′′ and {un} is a (C¯)c sequence for -Jλ, then {un} is bounded in X.Proof.
By1+un-Jλ′unX′→0, we have
(55)-Jλ′un,Qun=o1,-Jλ′un,Pun=o1.
Choosing v=Qun and v=Pun in (25), (55) implies that
(56)Quk2=∫R3‍unϕun·Qundx-∫R3‍λunp-2un-unq-2un·Qundx+o1,Pun2=-∫R3‍unϕun·Pundx+∫R3‍λunp-2un-unq-2un·Pundx+o1.
By these two equalities and un2=Pun2+Qun2 (see (23)), we obtain
(57)un2=∫R3‍un·ϕun·(Qun-Pun)dx-∫R3‍λunp-2un-unq-2un·Qun-Pundx+o1.
Let(58)χn,1x=1,ifunx≤D,0,ifunx>D
and χn,2=1-χn,1, where D comes from (50). Then, un2=(unχn,1)2+(unχn,2)2 and
(59)ϕun=ϕunχn,1+ϕunχn,2.
Since ϕunχn,1 is a solution of the equation
(60)-Δϕ=4πunχn,12inR3,
by (unχn,1)2≤D2 in R3 and the standard elliptic estimates (see, [28, Theorem  8.17]), we get that there exists a positive constant C2 that is independent of n and y∈R3, such that, for any y∈R3,
(61)ϕunχn,1L∞(B1(y))≤C2∫B2y‍unχn,14dx1/2≤C2D2∫B2y‍dx1/2.
Let C3=(∫B2(0)‍dx)1/2. By (61),
(62)ϕunχn,1L∞(R3)≤C2C3D2.
Together with ϕunχn,1≥0 in R3 (see Lemma 2(ii)), this implies
(63)0≤ϕunχn,1≤C2C3D2inR3.
By (57), (59), and (63), we have
(64)un2=∫R3‍un·ϕunχn,1·Qun-Pundx+∫R3‍un·ϕunχn,2·Qun-Pundx-∫{x∣unx≤D}‍λunp-2un-unq-2un·Qun-Pundx-∫{x∣|un(x)|>D}‍λunp-2un-unq-2un·Qun-Pundx+o1≤ϕunχn,1unL2(R3)Qun-PunL2(R3)+∫R3‍un·ϕunχn,2·Qun-Pundx+∫{x∣|un(x)|≤D}‍λunp-2un-unq-2un2dx1/2·Qun-PunL2(R3)+λ∫{x∣|un(x)|>D}‍unpdx(p-1)/p·Qun-PunLp(R3)+∫{x∣|un(x)|>D}‍unqdx(q-1)/q·Qun-PunLq(R3).
Since Qun-PunL2(R3)≤2C′un (C′ is the constant coming from (37)), Qun-PunLp(R3)≤C′′un, and Qun-PunLq(R3)≤C′′un, where C′′ is a positive constant independent of n and λ, we have
(65)un2≤2C′unϕunχn,1L2R3un+∫R3‍un·ϕunχn,2·Qun-Pundx+2C′∫x∣unx≤Dλunp-2unxxxxxxxxxxxxxxxxx-unq-2un2dx∫x∣unx≤D‍1/2un+λC′′∫x∣unx>D‍unpdxp-1/pun+C′′∫x∣unx>D‍unqdx(q-1)/qun.
By Lemma 7, we deduce that ∫{x∣unx>D}‍unpdx=o1 and ∫{x∣unx>D}‍unqdx=o1 if λ≤1. Together with (63), this implies that, for 0<λ≤1,
(66)un2≤2C′DC2C31/2unϕunχn,11/2L2(R3)un+∫R3‍un·ϕunχn,2·Qun-Pundx+2C′∫x∣unx≤Dλunp-2unxxxxxxxxxxxxxxxxxxx-unq-2un2dx∫x∣unx≤D1/2un+o1·un.
Using the Hardy-Littlewood-Sobolev inequality (see, (1.1) of [26]), we get that
(67)∫R3‍un·ϕunχn,2·Qun-Pundx=∫R3‍∫R3‍unχn,2y2·unx·Qun-Punxx-ydxdy≤CHLSunχn,22L6/5(R3)un·Qun-PunL6/5(R3)≤CHLSunχn,22L6/5(R3)unL12/5(R3)·Qun-PunL12/5(R3)≤C′′′∫{x∣unx>D}‍un12/5dx5/6un2,
where CHLS and C′′′ are positive constants independent of n and λ. By Lemma 7, we have
(68)∫{x∣unx>D}‍un12/5dx=o1if0<λ≤1.
Combining (68) and (67) with (66) yields that, for 0<λ≤1,
(69)un2≤2C′DC2C31/2unϕunχn,11/2L2R3un+o1·un2+2C′∫{x∣unx≤D}λunp-2unxxxxxxxxxxxxxxxxxx-unq-2un2dx∫{x∣unx≤D}1/2un+o1·un.
Let 0<ϵ<D. Then, for 0<λ≤1,
(70)∫{x∣unx≤ϵ}‍λunp-2un-unq-2un2dx1/2≤∫{x∣unx≤ϵ}‍unp-1+unq-12dx1/2≤2∫{x∣unx≤ϵ}‍un2p-2+un2q-2dx1/2≤4ϵp-2+ϵq-2∫{x∣unx≤ϵ}‍un2dx1/2≤4C′ϵp-2+ϵq-2un.
Combining (70) with (69) yields that, for 0<λ≤1,
(71)un2≤2C′DC2C31/2unϕunχn,11/2L2(R3)un+o1·un2+4C′2ϵp-2+ϵq-2un2+2C′∫{x∣ϵ<unx≤D}λunp-2unxxxxxxxxxxxxxxxxxxx-unq-2un2dx∫{x∣ϵ<unx≤D}1/2un+o1·un.
Let ϵ0∈(0,min⁡{D,D1/(q-p),1}) be such that 4C′2(ϵ0p-2+ϵ0q-2)<1/2. Then, (71) implies that
(72)un2≤4C′DC2C31/2unϕunχn,11/2L2(R3)un+o1·un2+o1·un+4C′∫{x∣ϵ0<unx≤D}λunp-2unxxxxxxxxxxxxxxxxxxx-unq-2un2dx∫{x∣ε0<unx≤D}1/2un.
Fromsup⁡n-Jλun≤c and (1+un)-Jλ′(un)X′→0, we obtain
(73)o1+c≥-Jλun+12Jλ′un,un=14∫R3‍un2ϕundx-∫R3‍λ12-1punp-12-1qunqdx.
It follows that
(74)14∫R3‍un2ϕundx+∫{x∣unx>λ1/(q-p)}12-1qunxq-λ12-1punpdx≤c+∫{x∣unx≤λ1/(q-p)}λ12-1punp-12-1qunqdx+o1≤c+C4λ(q-2)/(q-p)∫R3‍un2dx+o1,
where C4 is a positive constant depending only on p and q. Because (1/2-1/q)tq-λ(1/2-1/p)tp>0 if t≥λ1/(q-p), we get that, for 0<λ≤ϵ0q-p,
(75)∫{x∣D≥unx>ϵ0}‍12-1qunq-λ12-1punpdx≤∫{x∣unx>λ1/(q-p)}‍12-1qunq-λ12-1punpdx.
Then, by (74) and ∫R3‍un2ϕundx≥∫R3‍un2ϕunχn,1dx=unϕunχn,11/2L2(R3)2, we get that, for 0<λ≤ϵ0q-p,
(76)14unϕunχn,11/2L2(R3)2+∫{x∣D≥unx>ϵ0}12-1qunqxxxxxxxxxxxxxxxxxx-λ12-1punpdx≤c+C4λ(q-2)/(q-p)∫R3‍un2dx+o1≤c+C′2C4λ(q-2)/(q-p)un2+o1.
It is easy to verify that there exists a positive constant C5 depending only on p,q,D, and ϵ0 such that, if 0<λ≤ϵ0q-p/2 and ϵ0≤|t|≤D, then
(77)C5λtp-2t-tq-2t2≤12-1qtq-λ12-1ptp.
Together with (76), this yields that, for 0<λ≤ϵ0q-p/2,
(78)14unϕunχn,11/2L2(R3)2+C5∫{x∣D≥unx>ϵ0}‍λunp-2un-unq-2un2dx≤c+C′2C4λq-2/q-pun2+o1.
Combining (78) with (72), we deduce that there exists λ0′′∈(0,ϵ0q-p/2) that is small enough such that {un} is bounded if 0<λ<λ0′′.


## 5. Proof of Theorem1

Proof of Theorem1.
Letλ0=min⁡{λ0′,λ0′′}, where λ0′ and λ0′′ are the constants in Lemmas 5 and 8, respectively. By Lemmas 5 and 8, and Theorem  1.3 of [24], we deduce that, for any 0<λ<λ0, there exists a bounded (C¯)c sequence {un} for -Jλ with inf⁡nun>0. Up to a subsequence, either(a)
lim⁡n→∞sup⁡y∈R3∫B1(y)‍|un|2dx=0, or(b)
there existϱ>0 and an∈Z3 such that ∫B1(an)‍|un|2dx≥ϱ.
If (a) occurs, the Lions lemma (see [23, Theorem  1.21]) implies that un satisfies un→0 in Ls(R3) for any 2<s<6. It follows that
(79)∫R3‍λunp-2un-unq-2un·Qundx⟶0,∫R3‍λunp-2un-unq-2un·Pundx⟶0.
As (67), we have
(80)∫R3‍un·ϕun·Qun-Pundx=∫R3‍∫R3‍un2y·unx·Qun-Punxx-ydxdy≤CHLSun2L6/5(R3)un·Qun-PunL6/5(R3)≤CHLSun2L6/5(R3)unL12/5(R3)Qun-PunL12/5(R3)=CHLSunL12/5(R3)3Qun-PunL12/5(R3)⟶0,n⟶∞.
Combining (79) and (80) with (57) yields un→0. This contradicts inf⁡nun>0. Therefore, case (a) cannot occur. As case (b) therefore occurs, wn=un(·+an) satisfies wn⇀u0≠0. From (1+wn)-Jλ′(wn)X′=(1+un)-Jλ′(un)X′→0 and the weakly sequential continuity of -Jλ′, we have that -Jλ′(u0)=0. Therefore, u0 is a nontrivial solution of (1). This completes the proof.



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*Source: 652407-2015-01-14.xml*