# Existence of Multiple Nontrivial Solutions for a Strongly Indefinite Schrödinger-Poisson System

Authors: Shaowei Chen; Liqin Xiao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider a Schrödinger-Poisson system inℝ3 with a strongly indefinite potential and a general nonlinearity. Its variational functional does not satisfy the global linking geometry. We obtain a nontrivial solution and, in case of odd nonlinearity, infinitely many solutions using the local linking and improved fountain theorems, respectively.


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## Body

## 1. Introduction and Statement of Results

In this paper, we consider the Schrödinger-Poisson system:(1)-Δu+V(x)u+K1(x)ϕu=Q(x)f(u),inℝ3,-Δϕ=K2(x)u2,inℝ3.ForV, K1, K2, Q, and f, we assume the following.(v)
V∈Lloc⁡q(ℝ3) and V-=min⁡{-V,0}∈L∞(ℝ3)+Lq(ℝ3) for some q∈[2,∞). This assumption ensures that the Schrödinger operator S=-Δ+V is self-adjoint and semibounded on L2(ℝ3) (see Theorem A.2.7 in [1]). σ(S) denotes the spectrum of S. We assume that 0 lies in a gap of σ(S); that is, there exist α,β>0 such that
(2)σ(S)∩(-α,β)=∅.(k)
Q,Ki∈C(ℝ3) satisfy Q(x)>0,Ki(x)>0 for all x,
(3)lim⁡|x|→∞Q(x)=0,lim⁡|x|→∞Ki(x)=0,i=1,2,and there existsp∈(4,6) such that
(4)∫ℝ3Ki6p/(5p-12)(x)Q12/(5p-12)(x)dx<∞,i=1,2.(f1)
f∈C(ℝ) and there exists C>0 such that for all t∈ℝ,
(5)|f(t)|≤C(|t|+|t|p-1).(
f
2
)
f
(
t
)
=
o
(
t
) as t→0.
(
f
3
)
Let(6)F(t)=∫0t‍f(τ)dτ.
We assume that there existsA>0 such that for all t∈ℝ,
(7)F~(t):=tf(t)-2F(t)≥A|t|p.
(
f
4
)
f
(
-
t
)
=
-
f
(
t
) for all t∈ℝ.Remark 1.
Note that(8)(F(t)t2)′=tf(t)-2F(t)t3.
This together with (7) implies that there exists C1>0 such that for all t∈ℝ,
(9)F(t)≥C1|t|p.Our main results are as follows.Theorem 2.
Suppose that(v), (k), and (f1)–(f3) are satisfied, then the problem (1) has a nontrivial solution.Theorem 3.
Suppose that(v), (k), and (f1)–(f4) are satisfied, then the problem (1) has infinitely many solutions.Problem (1) arises in quantum mechanics and is related to the study of the nonlinear Schrödinger equation for a particle in an electromagnetic field or the Hartree-Fock equation. For a more detailed physical background of the Schrödinger-Poisson system, readers can refer to [2, 3] and the references therein.This system has attracted considerable research attention in the recent decade, and it has been studied widely by using the modern variational method and critical point theory under various assumptions. However, many mathematical studies have been devoted to the caseinf⁡ℝ3V>0. In this case, there are many results on the existence, nonexistence, or multiplicity of solutions for (1). One can refer to [2–20].There are very few studies devoted to (1) under the assumption that -Δ+V has a nontrivial negative eigenspace compared to the case inf⁡ℝ3V>0. In a recent paper [21], Chen and Liu studied the problem under the assumption on V that(v′)
V∈C(ℝ3) is bounded from below and μ(V-1(-∞,M])<∞ for every M>0, where μ is the Lebesgue measure on ℝ3. Moreover, the operator -Δ+V has negative eigenvalues.They verified the existence of multiple solutions of (1) under this assumption on V and under certain 4-superlinear conditions on f. Our assumptions on V, which are different from (v′), allow an infinitely dimensional negative eigenspace of -Δ+V. This causes some difficulties. For example, it makes the verification of the compactness conditions a more delicate problem. In addition, when we search for infinitely many solutions of (1) for the case where f is odd, the classical fountain theorem of Bartsch (see [22] or [23]) cannot be applied. Fortunately, this difficulty can be overcome using a recently improved fountain theorem of Batkama and Colin [24]. To the best of our knowledge, the Schrödinger-Poisson equation with a strongly indefinite linear part has never been studied. Besides the difficulties caused by the strongly indefinite linear part, the functional related to (1) (see Section 2) involves a nonlocal term and it makes the functional not satisfy the global linking structure. To overcome this difficulty and obtain a nontrivial solution of (1), we use the local linking method (see [25]).Throughout this paper, we denote the strong and the weak convergence by→ and ⇀, respectively. Lp(ℝ3) denotes the standard Lebesgue space with norm ∥u∥Lp=(∫ℝ3‍|u|pdx)1/p. For k∈ℕ, Hk(ℝ3) denotes the standard Sobolev space with norm ∥u∥Hk=(∫ℝ3‍∑|γ|≤k‍|∂γu|2dx)1/2. For a Banach space E, we denote the dual space of E by E*, and the norm of E* is denoted by∥·∥E*.


## 2. Proof of Theorem2

Assume that(v) holds and let S=-Δ+V be the self-adjoint operator acting on L2(ℝ3) with domain D(S)=H2(ℝ3). By virtue of (v), we have the orthogonal decomposition
(10)L2=L2(ℝ3)=L++L-
such that S is negative (resp. positive) in L-(resp. in L+). Let X=D(|S|1/2) be equipped with the inner product
(11)(u,v)=(|S|1/2u,|S|1/2v)L2
and norm ∥u∥=∥|S|1/2u∥L2, where (·,·)L2 denotes the inner product of L2. From (v),
(12)X=H1(ℝ3)
with equivalent norms. Therefore, X continuously embeds in Lq(ℝ3) for all 2≤q≤6. In addition, we have the decomposition
(13)X=X++X-,
where X±=X∩L± is orthogonal with respect to both (·,·)L2 and (·,·). Therefore, for every u∈X, there is a unique decomposition
(14)u=u++u-,u±∈X±
with (u+,u-)=0 and
(15)∫ℝ3‍|∇u|2dx+∫ℝ3‍V(x)u2dx=∥u+∥2-∥u-∥2,u∈X.Foru∈X, it is well known (see, e.g., Theorem 2.2.1 of [26]) that the Poisson equation
(16)-Δϕ=K2(x)u2
has a unique solution
(17)ϕu(x)=14π∫ℝ3‍K2(y)u2(y)|x-y|dy,
and ϕu∈𝒟1,2(ℝ3)={u∈L6(ℝ3)∣∫ℝ3|∇u|2dx<∞}.Let(18)Φ(u)=12∥u+∥2-12∥u-∥2+116π∬ℝ3K1(x)K2(y)u2(x)u2(y)|x-y|dxdy-∫ℝ3‍Q(x)F(u)dx,u∈X.
Under the assumptions (f1) and (k); Φ is a C1 functional in X. The derivative of Φ is given by
(19)〈Φ′(u),v〉=(u+,v)-(u-,v)+18π∬ℝ3K1(x)K2(y)u2(x)u(y)v(y)|x-y|dxdy+18π∬ℝ3K1(x)K2(y)u(x)v(x)u2(y)|x-y|dxdy-∫ℝ3‍Q(x)f(u)vdx,∀u,v∈X.
It is easy to see that if u is a critical point of Φ, then (u,ϕu) is a solution of (1).Our functionalΦ does not satisfy the geometric assumptions of the generalized linking theorem (see, e.g., [23, Chapter 6]) because of the term
(20)116π∬ℝ3K1(x)K2(y)u2(x)u2(y)|x-y|dxdy.
To overcome this difficulty, we apply the local linking theorem to find critical points of Φ.Recall that by definition (see [25]), a functional Ψ defined in X has a local linking at 0 with respect to the direct sum decomposition X=X-⊕X+, if there is ρ>0 such that
(21)Ψ(u)≤0,foru∈X-,∥u∥≤ρ,Ψ(u)≥0,foru∈X+,∥u∥≤ρ.
Let {ek±} be the total orthonormal sequences in X±. We consider two sequences of finite dimensional subspaces
(22)X1±⊂⋯⊂Xn±⊂⋯⊂X±,
where Xn±=span⁡{e1±,…,en±}. It is easy to see that
(23)X±=⋃n∈ℕ‍Xn±¯.
For n∈ℕ, let Xn=Xn-⊕Xn+ and Ψn denote the restriction of Ψ on Xn.Definition 4.
We say thatΨ∈C1(X) satisfies (C)* condition if any sequence {un}⊂X such that
(24)un∈Xn,sup⁡nΨ(un)<∞,(1+∥un∥)∥Ψn′(un)∥Xn*⟶0
contains a subsequence that converges to a critical point of Ψ.From [27, Theorem 2.2], we have the following.Theorem 5.
Suppose thatΨ∈C1(X) has a local linking at 0, Φ satisfies (C)* condition, Ψ maps bounded sets into bounded sets, and for every m∈ℕ,
(25)Ψ(u)⟶-∞,as∥u∥⟶∞,u∈X-⊕Xm+.
Then, Ψ has a nontrivial critical point.Lemma 6.
The functionalΦ has a local linking in 0 with respect to the direct sum decomposition X=X+⊕X-.Proof.
From the Hardy-littlewood-Sobolev inequality (see, e.g., [28]), we infer that there exists C>0 such that for every u∈X,
(26)∬ℝ3K1(x)K2(y)u2(x)u2(y)|x-y|dxdy≤C∥K1u2∥L6/5(ℝ3)∥K2u2∥L6/5(ℝ3)=C∥K11/2u∥L12/5(ℝ3)2∥K21/2u∥L12/5(ℝ3)2.
From (k), we deduce that Ki is bounded in ℝ3, i=1,2. Therefore, by the Sobolev inequality, we have
(27)∥Ki1/2u∥L12/5(ℝ3)≤C∥u∥L12/5(ℝ3)≤C∥u∥,i=1,2.
It follows that
(28)∬ℝ3K1(x)K2(y)u2(x)u2(y)|x-y|dxdy≤C∥u∥4
for some C>0.
From(f1) and (f2), we deduce that for any ϵ>0, there exists Cϵ>0 such that
(29)|F(t)|≤ϵt2+Cϵ|t|p,∀t∈ℝ.
This together with the fact that Q is a bounded function in ℝ3 (see (k)) implies that there exists C′>0 such that
(30)|∫ℝ3Q(x)F(u)dx|≤C′ϵ∥u∥2+C′Cϵ∥u∥p.
Combining (28), (30), and the definition of Φ (see (18)), we get that for any u∈X+,
(31)Φ(u)≥12∥u∥2-C′ϵ∥u∥2-C′Cϵ∥u∥p=(12-C′ϵ)∥u∥2-C′Cϵ∥u∥p,
and for any u∈X-,
(32)Φ(u)≤-12∥u∥2+C16π∥u∥4-C′ϵ∥u∥2-C′Cϵ∥u∥p=-(12-C′ε)∥u∥2+C16π∥u∥4+C′Cϵ∥u∥p.
Choose ϵ=1/4C′. Then, from the above two inequalities, we deduce that we can choose small ρ>0 such that Φ satisfies (21). Therefore, Φ has a local linking in 0 with respect to the direct sum decomposition X=X+⊕X-.Lemma 7.
Under the assumptions(v),(k), and (f1)–(f3), the functional Φ satisfies the (C)* condition.Proof .
Let{un} be a (C)c* sequence; that is, sup⁡nΦ(un)≤c and (1+∥un∥)∥Φ′(un)∥Xn*→0.
First, we prove that{un} is bounded in X. From (7), we have
(33)o(1)+2c≥2Φ(un)-〈Φ′(un),un〉=-18π∬ℝ3K1(x)K2(y)un2(x)un2(y)|x-y|dxdy+∫ℝ3‍Q(x)(unf(un)-2F(un))dx≥-18π∬ℝ3K1(x)K2(y)un2(x)un2(y)|x-y|dxdy+A∫ℝ3‍Q(x)|un|pdx.
From (26), we have
(34)∬ℝ3K1(x)K2(y)un2(x)un2(y)|x-y|dxdy≤C∥K1u2∥L6/5(ℝ3)∥K2u2∥L6/5(ℝ3).
By the Hölder inequality and (4), we have
(35)∫ℝ3‍Ki6/5(x)|un(x)|12/5dx≤(∫ℝ3‍Q(x)|un(x)|pdx)12/5p×(∫ℝ3‍Ki6p/(5p-12)(x)Q-12/(5p-12)(x)dx)1-(12/5p)≤C(∫ℝ3‍Q(x)|un(x)|pdx)12/5p,i=1,2.
Combining (34) and (35), we get that
(36)∬ℝ3K1(x)K2(y)un2(x)un2(y)|x-y|dxdy≤C(∫ℝ3‍Q(x)|un(x)|pdx)4/p.
This together with (33) yields that ∫ℝ3‍Q(x)|un(x)|pdx is bounded.
Second, we prove that∥un∥ is bounded. We have
(37)o(1)=〈Φ′(un),un+-un-〉=∥un∥2-∫ℝ3Q(x)f(un)(un+-un-)dx+18π∬ℝ3((|x-y|)-1((un+(y)-un-(y))K1(x)K2(y)un2(x)un(y)hhhhhhhhhhhhll×(un+(y)-un-(y))K1(x)K2(y)un2(x)un(y))hhhhhhhlhhlhhll×(|x-y|)-1)dxdy+18π∬ℝ3((|x-y|)-1((un+(x)-un-(x))un2(y)K1(x)K2(y)un(x)hhhhhhhhhhllhh×(un+(x)-un-(x))un2(y))hhhhhhhhlhlhhll×(|x-y|)-1)dxdy.
From (5) and the Hölder inequality, we get that for any R>0,(38)∫ℝ3‍Q(x)f(un)(un+-un-)dx≤C∫ℝ3‍Q(x)|un|·|un+-un-|dx+C∫ℝ3‍Q(x)|un|p-1|un+-un-|dx≤C∫{x∈ℝ3∣|x|>R}‍Q(x)|un|·|un+-un-|dx+C∫{x∈ℝ3∣|x|≤R}‍Q(x)|un|·|un+-un-|dx+C(∫ℝ3‍Q(x)|un|pdx)(p-1)/p×(∫ℝ3‍Q(x)|un+-un-|pdx)1/p.
Since ∫ℝ3‍Q(x)|un|pdx is bounded,
(39)C(∫ℝ3‍Q(x)|un|pdx)(p-1)/p(∫ℝ3‍Q(x)|un+-un-|pdx)1/p≤C∥un∥.
From lim⁡|x|→∞Q(x)=0 (see (k)) and the Sobolev inequality, we get that there exists R>0 such that
(40)C∫{x∈ℝ3||x|>R}‍Q(x)|un|·|un+-un-|dx≤Csup⁡|x|>RQ·∫{x∈ℝ3∣|x|>R}‍|un|·|un+-un-|dx≤Csup⁡|x|>RQ·∥un∥2≤12∥un∥2.
From the Hölder and Sobolev inequalities and boundedness of ∫ℝ3‍Q(x)|un|pdx, we have
(41)C∫{x∈ℝ3∣|x|≤R}‍Q(x)|un|·|un+-un-|dx≤C(∫{x∈ℝ3||x|≤R}Q(x)‍dx)p/(p-2)(∫ℝ3Q(x)|un|pdx)1/p×(∫ℝ3‍Q(x)|un+-un-|pdx)1/p≤DR∥un∥,
where DR is a positive constant that depends only on R. Combining (38)–(41), we infer that there exists R>0 such that
(42)∫ℝ3‍Q(x)f(un)(un+-un-)dx≤12∥un∥2+(DR+C)∥un∥.
From the Hardy-Littlewood-Sobolev and Hölder inequalities, we get that(43)∬ℝ3K1(x)K2(y)un2(x)un(y)(un+(y)-un-(y))|x-y|dxdy≤C∥K1un2∥L6/5(ℝ3)∥K2un(un+-un-)∥L6/5(ℝ3)≤C∥K11/2un∥L12/5(ℝ3)2∥K21/2un∥L12/5(ℝ3)×∥K21/2(un+-un-)∥L12/5(ℝ3).
From (43), (35), and the boundedness of ∫ℝ3‍Q(x)|un|pdx, we get that there exists a constant C>0 such that
(44)∬ℝ3K1(x)K2(y)un2(x)un(y)(un+(y)-un-(y))|x-y|dxdy≤C∥K21/2(un+-un-)∥L12/5(ℝ3)≤C∥un∥.
Similarly, we have
(45)∬ℝ3K1(x)K2(y)un(x)(un+(x)-un-(x))un2(y)|x-y|dxdy≤C∥un∥.
Combining (37), (42), (44), and (45), we infer that ∥un∥ is bounded.
Finally, we prove that{un} has a convergent subsequence. Up to a subsequence, we assume that un⇀u in X. Then, u is a critical point of Φ. From (f1),
(46)∫ℝ3‍Q(x)f(un)u+dx⟶∫ℝ3‍Q(x)f(u)u+dx
and from lim⁡|x|→∞Q(x)=0 and un⇀u,
(47)|∫ℝ3‍Q(x)f(un)(un+-u+)dx|≤C∫ℝ3‍Q(x)|un|p-1|un+-u+|dx⟶0.
Therefore,
(48)∫ℝ3‍Q(x)f(un)un+dx-∫ℝ3‍Q(x)f(u)u+dx⟶0.
Byun⇀u and lim⁡|x|→∞Ki(x)=0,i=1,2, we get that
(49)|∬ℝ3K1(x)K2(y)un2(x)un(y)un+(y)|x-y|dxdy-∬ℝ3K1(x)K2(y)u2(x)u(y)u+(y)|x-y|dxdy|≤∥K1(un2-u2)∥L6/5(ℝ3)∥K2(unun+-uu+)∥L6/5(ℝ3)⟶0.
Similarly,
(50)|∬ℝ3K1(x)K2(y)un(x)un+(x)un2(y)|x-y|dxdy-∬ℝ3K1(x)K2(y)u(x)u+(x)u2(y)|x-y|dxdy|⟶0.
By (48)–(50), 〈Φ′(u),u+〉=0, and
(51)o(1)=〈Φ′(un),un+〉kkkl=∥un+∥2-∫ℝ3‍Q(x)f(un)un+dx+18π∬ℝ3K1(x)K2(y)un2(x)un(y)un+(y)|x-y|dxdy+18π∬ℝ3K1(x)K2(y)un(x)un+(x)un2(y)|x-y|dxdy,
we get that un+→u+ in X. The same argument implies that un-→u- in X. Therefore, un→u in X.Remark 8.
From the proof of this theorem, we infer thatΦ also satisfies the Cerami condition; that is, if {un}⊂X satisfies
(52)sup⁡nΦ(un)<+∞,(1+∥un∥)∥Φ′(un)∥X*⟶0,
then {un} contains a convergent subsequence.Lemma 9.
The functionalΦ satisfies (25).Proof.
If the functional does not satisfy (25), then there exist M>0,m∈ℕ, and un∈Xm+⊕X- such that ∥un∥→∞ and
(53)Φ(un)=12∥un+∥2-12∥un-∥2+116π∬ℝ3K1(x)K2(y)un2(x)un2(y)|x-y|dxdy-∫ℝ3‍Q(x)F(un)dx≥-M.
This together with (36) and (9) yields
(54)12∥un+∥2-12∥un-∥2+C(∫ℝ3‍Q(x)|un(x)|pdx)4/p-C1∫ℝ3‍Q(x)|un(x)|pdx≥-M.
Since p>4, there exists D>0 such that
(55)D-C12∫ℝ3‍Q(x)|un(x)|pdx≥C(∫ℝ3‍Q(x)|un(x)|pdx)4/p-C1∫ℝ3‍Q(x)|un(x)|pdx.
Then, by (54),
(56)12∥un+∥2-12∥un-∥2-C12∫ℝ3‍Q(x)|un(x)|pdx≥-M-D.
We use E to denote the closure of Xm+⊕X- under the norm ||u||Q,p=(∫ℝ3‍Q(x)|u(x)|pdx)1/p. Since there exists a continuous projection P:E→Xm+, there exists C3>0 such that for every u∈Xm+⊕X-,
(57)(∫ℝ3‍Q(x)|u(x)|pdx)1/p≥C3(∫ℝ3‍Q(x)|u+(x)|pdx)1/p.
Since ∥un∥→∞ and Xm+ is finite dimensional, from (56) and (57), we get that
(58)-M-D≤12∥un+∥2-12∥un-∥2-C1C3p2∫ℝ3‍Q(x)|un+(x)|pdx⟶-∞.
This contradiction implies that Φ satisfies (25).Proof of Theorem2.
The desired result of Theorem2 can be derived by combining Theorem 5 and Lemmas 6, 7, and 9.


## 3. Proof of Theorem3

Recall that{ek±} are the total orthonormal sequences in X±. For k=1,2,…, let
(59)Yk=X-⊕Xk+=X-⊕span⁡{e1+,…,ek+},Zk=span⁡{ek+,ek+1+,…}¯.LetP:X→X-, Q:X→X+ be the orthogonal projections. As [23], on X, we define
(60)|∥u∥|=max⁡{∥Qu∥,∑j=1∞‍12k+1|(Pu,ek-)|}.
The topology generated by |∥·∥| will be denoted by τ and all topological notations related to it will include this symbol.For the proof of Theorem3, we use the following improved fountain theorem of Batkam and Colin [24, Theorem 12], which is a generalization of the classical fountain theorem of Bartsch [22] (see also [23]).Theorem 10.
Assume that an even functionalΨ∈C1(X) satisfies the following:(A)
Ψ is τ-upper semicontinuous, and the gradient ∇Ψ is weakly sequentially continuous.If there exist ρk>rk>0 such that(i)
bk=inf⁡u∈Zk,∥u∥=rkΨ(u)→+∞, as k→∞,(ii)
ak=max⁡u∈Yk,∥u∥=ρkΨ(u)≤0, then there exist {ck}⊂ℝ and sequences {uk,n}⊂X such that for every k∈ℕ, ck≥bk and
(61)Ψ(uk,n)⟶ck,(1+∥uk,n∥)∥Ψ′(uk,n)∥X*⟶0,asn⟶∞.Remark 11.
In [24], the result of this fountain theorem is
(62)Ψ(uk,n)⟶ck,∥Ψ′(uk,n)∥X*⟶0,asn⟶∞.
Since the deformation theorem is still valid under the Cerami condition (see, e.g., [27]), replacing the pseudogradient vector field 2∥∇Ψ(v)∥-2∇Ψ(v) in page 442 of [24] by the Cerami-type pseudogradient vector field (1+∥v∥)∇Ψ(v), we see that similar to many critical point theorems, the result of the fountain theorem in [24] can be improved as (61).Proof of Theorem3.
(1) Let us prove thatΦ is τ-upper semicontinuous. Assume that |∥un-u∥|→0 and c≤Φ(un). Since un+=Qun→Qu, un-=Pun are bounded so that Pun⇀Pu, it follows that un⇀u in X. Since lim⁡|x|→∞Q(x)=0, from (f1) and un⇀u, we obtain
(63)∫ℝ3‍Q(x)F(un)dx⟶∫ℝ3‍Q(x)F(u)dx.
In addition, from the proof of (49) we infer that
(64)|∬ℝ3K1(x)K2(y)un2(x)un2(y)|x-y|dxdy-∬ℝ3K1(x)K2(y)u2(x)u2(y)|x-y|dxdy|⟶0.
Using the Fatou lemma, we obtain
(65)-Φ(u)=∥Pu∥22-∥Qu∥22-116π×∬ℝ3K1(x)K2(y)u2(x)u2(y)|x-y|dxdy+∫ℝ3Q(x)F(u)dx≤liminf⁡n→∞(∥Pun∥22-∥Qun∥22kkkkkkl-116π∬ℝ3K1(x)K2(y)un2(x)un2(y)|x-y|dxdykkkkkkl+∫ℝ3Q(x)F(un)dx∥Pun∥22)=liminf⁡n→∞-Φ(un)≤-c.
(2) The proof that∇Ψ is weakly sequentially continuous is similar to that in the proof of Lemma 6.15 of [23].
(3) Verification of (i) forΦ: since lim⁡|x|→∞Q(x)=0, we infer that if uk⇀0 in X, then
(66)∫ℝ3‍Q(x)|un|pdx⟶0.
It follows that
(67)βk=sup⁡u∈Zk,∥u∥=1∥u∥Q,p⟶0,ask⟶∞,
where
(68)∥u∥Q,q=(∫ℝ3‍Q(x)|u|qdx)1/q.
From (f1) and (f2), we deduce that for any ϵ>0, there exists Cϵ>0 such that
(69)|F(t)|≤ϵ|t|2+Cϵ|t|p,∀t∈ℝ.
For u∈Zk, by the Sobolev inequality and (67),
(70)Φ(u)=12∥u∥2+116π∬ℝ3K1(x)K2(y)u2(x)u2(y)|x-y|dxdy-∫ℝ3‍Q(x)F(u)dx≥12∥u∥2-ϵ∥u∥Q,22-Cϵ∥u∥Q,pp≥12∥u∥2-Cϵ∥u∥2-Cϵβkp∥u∥p.
Choosing ϵ=1/4C and letting rk=(2Cϵpβkp)1/(2-p), we get from (70) that for ∥u∥=rk,
(71)Φ(u)≥12(12-1p)(2Cϵpβkp)2/(2-p).
Since βk→0 and p>2, it follows that
(72)bk=inf⁡u∈Zk,∥u∥=rkΦ(u)⟶+∞.
(4) Verification of (ii) forΦ: since Yk=X-⊕Xk+, (ii) is a direct consequence of Lemma 9.
Finally, from(f4), we obtain that Φ is an even functional. Then, from (1)–(4) and Remark 8, we deduce that for every k, there exists a critical point uk of Φ such that Φ(uk)≥bk. Therefore, (1) has infinitely many solutions {(uk,ϕuk)}.



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*Source: 240208-2014-02-19.xml*