R E S E A R C H
Open Access
Gradient systems with sublinear term near
the origin and asymptotically linear term near
infinity
Hongrui Cai
*and Jiabao Su
*Correspondence:
School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China
Abstract
In this paper we study the existence of nontrivial solutions for a sublinear gradient system with a nontrivial critical group at infinity.
MSC: 35J10; 35J65; 58E05
Keywords: gradient system; sublinear; critical group; Morse theory
1 Introduction
In this paper, we are concerned with the following gradient system
⎧ ⎪ ⎨ ⎪ ⎩
–u=Fu(x,u,v), x∈,
–v=Fv(x,u,v), x∈,
u=v= , x∈∂,
(GS)
where⊂RNis a bounded open domain with a smooth boundary∂andF
udesignates
the partial derivative with respect touof the nonlinearityF:×R→R. The solutions of such systems are steady-states of reaction-diffusion systems arising in many applied sciences such as biology, chemistry, ecology or physics. It is well known that (GS) has variational structure when the nonlinearityFsatisfies the subcritical growth condition
(F) F∈C(×R,R)and there areC> and <p< ∗such that ∇F(x,z)≤C +|z|p–, forx∈,z= (u,v)∈R,
where∗=NN– ifN≥and∗=∞ifN= , .
That is, the solutions of (GS) can be found as critical points of the following functional
(u,v) :=
|∇u|+|∇v|dx–
F(x,u,v)dx
defined onE:=H
()×H() which is a Hilbert space endowed with the inner product
z,w=(u,v), (φ,ψ)=
(∇u∇φ+∇v∇ψ)dx, z= (u,v),w= (φ,ψ)∈E
and the associated norm
z=
|∇z|dx=
|∇u|+|∇v|dx, z= (u,v)∈E.
By the compact Sobolev embeddingE→Lq()×Lq() withq∈[, ∗), under the global
assumption (F), the functionalis well defined and is of classC(see []) with its Fréchet derivative
(u,v), (ϕ,ψ)=
(∇u∇ϕ+∇v∇ψ)dx–
∇F(x,u,v), (ϕ,ψ)dx
for (u,v), (ϕ,ψ)∈E. The weak solutions to (GS) inEare exactly critical points ofinE. We make some conventions. We use | · |and (·,·) to denote the norm and the inner product inRand usez= (u,v) to denote an element inRandE.Bzdenotes the matrix product inR for a × matrixBandz= (u,v)∈R. We use to denote the origin in various spaces. LetM() be the set of all continuous, cooperative and symmetric matrix functions onR. A matrix functionA∈M
() takes the form
A(x) =
a(x) b(x) b(x) c(x)
with the functionsa,b,c∈C(,¯ R) satisfying the conditions thatb(x)≥ for allx∈, which meansAis cooperative, and thatmaxx∈ ¯{a,c}> .
WhenF satisfies ∇F(x, ) = ,F(x, ) = forx∈, the system (GS) admits a trivial solutionz= . We are interested in the nontrivial solutions for (GS). In the current paper we apply the Morse theory to study the existence of nontrivial solutions of (GS) when the problem issublinearnear the origin and isasymptotically linearnear infinity.
We make the following assumption near the origin.
(F) ∇F(x, ) = ,F(x, ) = and there areδ> and <σ< such that <∇F(x,z),z≤σF(x,z) forz∈Rwith <|z| ≤δ,x∈.
In order to state the assumptions on the nonlinearity at infinity, we need some basic facts about the eigenvalue problem of linear gradient system. For a given matrixA∈M(), it is known (see [, ]) that the corresponding linear system
–z=λA(x)z, x∈,
z= , x∈∂, (LA) admits a sequence of distinct eigenvalues of finite multiplicity
<λ(A) <λ(A) <· · ·<λk(A) <· · ·
such thatλk(A)→ ∞ask→ ∞. According toA, the spaceEcan be split as
where
EA=ker(+A), EA–=
λk(A)<
ker+λk(A)A
.
The numbersm(A) =dimEA–,n(A) =dimEAare well determined and finite.
We assume that the nonlinear system (GS) is asymptotically linear at infinity in the sense that the functionFsatisfies
(F∞) there is a matrixB∞∈M()such that
∇F(x,z) –B∞(x)z=o|z|, |z| → ∞,z∈R,x∈.
Associated toB∞, we setE–∞=E–B∞,E∞=EB∞,E+∞=E+B∞. Denotem∞=m(B∞) :=dimE∞–,
n∞=nB∞:=dimE∞. We say that the system (GS) is nonresonant at infinity ifn∞= , while
it is resonant at infinity ifn∞> .
We first consider the nonresonance case. We have the following.
Theorem . Assume that F satisfies(F), (F∞)and n∞= .Then(GS)has at least one nontrivial weak solution in E.
Next we consider the resonance case. We need additional assumptions onFnear infin-ity.
(F∞) λ(B∞) = and
lim
|z|→∞
F(x,z) – (B∞z,z)= –∞ uniformly in.
(F∞±) λk(B∞) = for somek≥. Forzn=yn+wn, whereyn∈E∞,wn∈E∞⊥,zn → ∞
and yn
zn→imply that there existδ> andN∈Nsuch that
±
∇F(x,zn) –B∞zn,yn
dx≥δ forn≥N.
Theorem . Let F satisfy(F), (F∞)and(F∞).Then (GS)has at least one nontrivial
weak solution in E.Moreover,if F is even in z,then(GS)has infinitely many nontrivial weak solutions in E.
Theorem . Let F satisfy(F), (F∞)and(F∞±).Then(GS)has at least one nontrivial weak solution in E.
Landesman-Lazer type resonance condition [] which can be formulated as
|∇F(x,z) –B∞(x)z| ≤C, x∈,z∈R,
limy→∞(F(x,y) – (B∞(x)y,y))dx=±∞, y∈E∞.
(LL±)
See [] for details. Near the origin we impose (F), which means that∇Fis sublinear orF is sub-quadratic near zero. This kind of condition caught our attention first in a preprint by Liu and Wu [] where a single elliptic equation was considered. This is the first use for gradient system in the current paper.
The asymptotically linear gradient systems (GS) have received some attention for years. We mention some recent related works [–] and the references therein. In these works, existence and multiplicity of nontrivial solutions for (GS) were obtained by combining var-ious arguments involving Morse theory, saddle point reduction method (see [–]) and three critical point theorem (see []),etc.All above mentioned works dealt with the case that at least one of the critical groups ofat isnontrivialsomewhere. In the present pa-per, we study via Morse theory the case that all critical groups ofat are trivial under the condition (F). Due to (F), the saddle point reduction methods [–] cannot be applied and there is no linking at . Comparing with known ones, the existence and multiplicity results for (GS) are all new. See more remarks in the last section of the paper.
The paper is organized as follows. In Section , we collect some basic abstract tools. In Section we compute the critical groups at zero and infinity. The proofs of Theorems .-. and comments are given in Section .
2 Preliminary
In this section we cite some preliminaries that will be used to prove the main results of the paper. We first collect some results on Morse theory (see [, ]) for aCfunctional defined on a Hilbert spaceE.
Let∈C(E,R). Denote forc∈R
c=z∈E:(z)≤c, Kc=
z∈E:(z) = ,(z) =c.
We say thatpossesses the deformation property at the levelc∈Rif for any¯> and any neighborhoodN ofKc, there are> and a continuous deformationη:E×[, ]→E
such that
() η(z,t) =zfor eithert= orz∈/–[c–¯,c+¯]; () (η(z,t))is non-increasing intfor anyz∈E; () η(c+\N)⊂c–.
We say thatpossesses the deformation property ifpossesses the deformation property at each levelc∈R.
In applications the deformation property is ensured by the Palais-Smale condition or the Cerami condition.
We say thatsatisfies the Palais-Smale condition at the levelc∈Rif any sequence
{zn} ⊂Esatisfying(zn)→cand(zn)→ asn→ ∞has a convergent subsequence.
con-vergent subsequence.satisfies the Cerami condition ifsatisfies the Cerami condition at eachc∈R.
Ifsatisfies the Palais-Smale condition or the Cerami condition, thenpossesses the deformation property [, ].
Letzbe an isolated critical point ofwith(z) =c∈R, andU be a neighborhood ofz. The group
Cq(,z) :=Hq
c∩U,c∩U\ {z}
, q∈Z
is called theqth critical group ofatz, whereH∗(A,B) denotes a singular relative
ho-mology group of the pair (A,B) with integer coefficients.
LetK={z∈E:(z) = }. Assume that(K) is bounded from below bya∈Rand possesses the deformation property at allc≤a. Then the group
Cq(,∞) :=Hq
E,a, q∈Z
is called theqth critical group ofat infinity [].
Assume thatsatisfies the deformation property andKis a finite set. The Morse type numbers of the pair (E,a) are defined byMq:=
z∈KdimCq(,z), and the Betti numbers
of the pair (E,a) are defined byβq:=dimCq(,∞).
Proposition . Assume that∈C(E,R)possesses the deformation property,#K<∞, and all Mq,βqare finite and only finitely many of them are nonzero.Then it holds
q
j=
(–)q–jMj≥ q
j=
(–)q–jβj (Morse inequality), (.)
∞
q=
(–)qMq= ∞
q=
(–)qβq (Morse equality). (.)
IfK=∅, thenβq= for allq∈Z. From (.) one can deduce thatMq≥βqfor allq∈Z.
Thus ifβq= for someq∗∈Z, thenmust have a critical pointz∗withCq∗(,z∗). If
K={z∗}, thenCq(,∞)∼=Cq(,z∗) for allq∈Z. Thus ifCq(,∞)Cq(,z∗) for some
q∈Z, thenmust have a new critical point. Therefore the basic idea in applying Morse theory to find critical points ofis to compute critical groups both at infinity and at known critical points clearly and then to find unknown critical points by applying formulas (.) and (.).
Now we state an abstract result for the critical groups at infinity.
Proposition . Let the functional:E→Rtake the form
(z) =
Lz,z+(z), (.)
whereL:E→E is a self-adjoint linear operator such thatis isolated inσ(L),the spec-trum ofL.Assume that∈C(E,R)satisfies
Denote V :=kerL,W:=V⊥=W+⊕W–,where W±are subspaces on whichLis positive (negative)definite.Assume thatμ=dimW–andν=dimV are finite,andpossesses the deformation property.
() Ifν= ,then
Cq(,∞)∼=δq,μZ, q∈Z. () Ifν> ,then
Cq(,∞)∼=δq,k±Z, k+=μ,k–=μ+ν
providedsatisfies the angle conditions with respect toE=V⊕W: (AC±
∞) there existM> and∈(, )such that
±(z),y≥, forz=y+w,z ≥M,w ≤z,y∈V,w∈W.
Proposition .() was obtained in [] (see Remark . in []). Proposition .() is a revision of Proposition . in [] which was made first in [] and was remade in [].
Next we recall an abstract critical point theorem built by Wang in [].
Proposition .([]) Let∈C(X,R),where X is a Banach space.Assume that
pos-sesses the deformation property,is even and bounded from below,and() = .If for any k∈N,there exist k-dimensional subspaces Xkandρk> such that
sup
z∈Xk∩Sρk
(z) < , (.)
where Sρ={u∈X| u=ρ}, thenhas a sequence of critical values ck< satisfying
ck→as k→ ∞.
Finally, we mention the eigenvalues of the linear gradient system (LA). By the compact embeddingE→L()×L(), for a givenA∈M(), there is a compact self-adjoint operatorTA:E→Eassociated withAsuch that
TAz,w=
A(x)z,wdx, z,w∈E.
The operatorTApossesses the property thatλ(A) is an eigenvalue of (LA) if and only if there is nonzeroz∈Esuch that
λ(A)TAz=z.
(LA) has a sequence of distinct eigenvalues
<λ(A) <λ(A) <· · ·<λk(A) <· · · → ∞,
negative, positive definite invariant subspaces and the kernel ofI–TA, respectively. We
refer to [, ] for more properties related to the eigenvalue problem (LA) and the opera-torTA.
3 Critical groups and compactness
In this section we verify the compactness of the functionaland compute the critical groups ofat both zero and infinity. Without loss of generality, we assume that (GS) has finitely many weak solutions so that the trivial solutionz= is an isolated critical point of. We first compute the critical groupsCq(, ). The idea was from an unpublished
preprint by Liu and Wu [] where a single elliptic equation was studied. We work with the functional
(z) =
|∇z|dx–
F(x,z)dx, z∈E.
Lemma . Assume that F satisfies(F)and(F),then
Cq(, )∼= for all q∈Z. (.)
Proof Denote Bρ :={z∈ E :z ≤ρ}. By definition of critical groups, we can write Cq(, )Hq(Bρ∩,Bρ∩\ {}). We will construct a deformation mapping from
(Bρ,Bρ\ {}) to (Bρ∩,Bρ∩\ {}) forρ> small.
In the following we usecito denote positive constants. By (F), one deduces that
F(x,z)≥c|z|σ forx∈,z∈R,|z| ≤δ. (.)
It follows from (F) and (.) that for somep∈(, ∗),
F(x,z)≥c|z|σ –c|z|p, x∈,z∈R. (.)
Forz∈Eands> , we have
(sz) = s
|∇z|dx–
F(x,sz)dx
≤
s z–
c|sz|σ –c|sz|p
dx
≤
s
z–c
sσzσLσ +cspzpLp. (.)
Sinceσ< <p, for each givenz∈E\ {}, there existss=s(z) > such that
Letz∈Ebe such thatz= and(z) = . Then d
ds(sz)|s==
|∇z|dx–
∇F(x,z),zdx
=
–σ
z–
∇F(x,z),z–σF(x,z)dx
≥
–σ
z–c
|z|pdx
≥
–σ
z–c
zp. (.)
From (.), one concludes that there existsρ> such that
d
ds(sz)|s=> forz∈Ewith(z) = and <z ≤ρ. (.) From now on we fixρ> . We claim that
z∈Bρand(z) < ⇒ (sz) < for alls∈(, ). (.)
Letz∈Bρand(z) < . By the continuity of, there existsτ ∈(, ] such that(sz) < for alls∈( –τ, ). We will get (.) by provingτ= . Suppose that <τ< . Then there is somes∈(, –τ] such that
(sz) = , (sz) < fors<s< .
Assz∈Bρ, it follows from (.) that d
ds
s(sz)
|s=> . (.)
But(sz) –(sz) =(sz) < implies that
d ds
s(sz)
|s=≤. (.)
This contradicts (.). Thusτ= and (.) holds. Now define a mappingπ:Bρ→[, ] as
π(z) =
forz∈Bρwith(z)≤,
s∈(, ) forz∈Bρwith(z) > and(sz) = . (.) By (.), (.) and (.), forz∈Bρwith(z) > , there exists a uniqueπ(z)∈(, ) such that
⎧ ⎪ ⎨ ⎪ ⎩
(π(z)z) = ,
(sz) < for alls∈(,π(z)), (sz) > for alls∈(π(z), ).
Thus the mappingπis well defined. Moreover, it follows from (.), (.) and the implicit function theorem that the mappingπis continuous inz. Define a mappingη: [, ]×Bρ→ Bρby
η(s,z) = ( –s)z+sπ(z)z, s∈[, ],z∈Bρ.
Then ηis a continuous deformation from (Bρ,Bρ\ {}) to (Bρ∩,Bρ∩\ {}). By homotopy invariance of a homology group and the contractibility ofBρ\ {}, we have
Cq(, ) =Hq
Bρ∩,Bρ∩\ {}∼=Hq
Bρ,Bρ\ {}∼= for allq∈Z.
The proof is complete.
We remark here that in [] the similar idea for computing the critical groups at was presented for a single elliptic equation. For (GS), the conditions used in [] can be for-mulated as
(F˜) there areδ> and <σ< such that
(i) F(x,z)≥c|z|σ forz∈Rwith|z| ≤δ,x∈, (ii) F(x,z) –∇F(x,z),z> for allz= ,x∈.
We note here that (F˜) is not comparable with (F) since (F) is a local condition and although (F) implies (F˜)(i) but (F˜)(ii) is a global condition.
Now we verify the compactness for the functionaland compute the critical groups of at infinity. To do this, we rewrite the functionalas
(z) =
|∇z|– (B∞z,z)dx–
G(x,z)dx, z∈E,
where G(x,z) = F(x,z) – (B∞z,z).
Lemma . Let F satisfy(F∞)and(F∞).
(i) The functionalis coercive onEand satisfies the Palais-Smale condition. (ii) Cq(,∞)∼=δq,Z,q∈Z.
Proof (i) First, (F∞) implies (F) while (F∞) implies
lim
|z|→∞G(x,z) = –∞ uniformly in. (.)
We will prove that
(z)→ ∞ asz → ∞,z∈E. (.)
Assume that there is a sequence{zn} ⊂Esuch that for someM> , it holds
zn → ∞ asn→ ∞. (.)
Setz˜n=zznn. Then˜zn= for alln∈N. Up to a subsequence, we may assume that there
is˜z∈Esuch that
⎧ ⎪ ⎨ ⎪ ⎩
˜
znz˜ weakly inE,
˜
zn→ ˜z strongly inLp()×Lp(), ≤p< ∗,
˜
zn(x)→ ˜z(x) for a.e.x∈.
(.)
By (.) one has that for some constantc> ,
G(x,z)≤c forx∈,z∈R. Therefore by (.) we deduce that
M
zn
≥(zn)
zn
≥
˜zn –
(B∞z˜n,z˜n)dx–
c||
zn
. (.)
Takingn→ ∞in (.), it follows from (.) and (.) that
lim sup
n→∞ ˜zn
≤
(B∞z˜,z˜)dx. (.)
On the other hand, we have by the lower semi-continuity of the norm that
(B∞˜z,˜z)dx≤ ˜z≤lim inf
n→∞ ˜zn
. (.)
Thus
lim
n→∞˜zn
=˜z
. (.)
By (.), (.) we get
˜
zn→ ˜zstrongly inE and ˜z=
(B∞˜z,˜z)dx. (.)
Hence˜z= andz˜is an eigenvector corresponding to the first eigenvalueλ(B∞) = . It follows thatz˜= for almost everyx∈and then
zn(x)→ ∞ for a.e.x∈. (.)
Now it follows from (.), (.) and the Fatou lemma that
M≥ zn
–
(B∞zn,zn)dx–
Gx,zn(x)
dx
≥–
Gx,zn(x)
dx→ ∞ asn→ ∞. (.)
By the coercivity of, a Palais-Smale sequence{zn}ofmust be bounded. SinceFhas
a subcritical growth, a standard argument shows that{zn}has a convergent subsequence.
(ii) Sinceis coercive and weakly lower semi-continuous,is bounded from below. Takea<infz∈E(z). Then
Cq(,∞) =Hq
E,a∼=Hq(E,∅)∼=δq,Z, q∈Z.
The proof is complete.
Lemma . Let F satisfy(F∞)and(F∞±). (i) satisfies the Cerami condition. (ii) Cq(,∞)∼=δq,m∞+n∞Zif(F∞+)holds. (iii) Cq(,∞)∼=δq,m∞Zif(F∞–)holds. Proof (i) Let{zn} ⊂Ebe such that
+zn(zn)→ asn→ ∞. (.)
We only need to show that{zn}is bounded inE. Suppose, by the way of contradiction, that
zn → ∞ asn→ ∞. (.)
Denotez¯n=zznn, then¯zn= . Passing to a subsequence if necessary, we may assume that
there is¯z∈Esuch that asn→ ∞,
⎧ ⎪ ⎨ ⎪ ⎩
¯
zn→ ¯z strongly inL()×L(),
¯
znz¯ weakly inE,
¯
zn(x)→ ¯z(x) almost everywhere in.
(.)
By (F∞) and (.) we deduce that{∇F(x,zn(x))–znB∞ (x)zn(x)}is bounded inL
()×L() and
∇F(x,zn) –B∞(x)zn
zn →
almost everywhere in.
Therefore
∇F(x,zn) –B∞(x)zn
zn
inL()×L(). (.)
Forw∈E, we have
(zn),w
zn
=
∇¯zn∇w–
B∞(x)z¯n,w
dx
–
∇F(x,zn) –B∞(x)zn
zn
,w
dx. (.)
Lettingn→ ∞, it follows that
Takingw=z¯nin (.) and using (.), (.), we obtain that
=
B∞(x)z,¯ z¯dx. (.)
Takingw=z¯in (.), we obtain that
¯z=
B∞(x)z,¯ z¯dx= . (.)
Therefore¯zn → ¯z= and thus
¯
zn→ ¯z strongly inE
andz¯is an eigenvector of (LB∞) associated with eigenvalue . It follows that¯z∈E∞. Write
zn=yn+wn, whereyn∈E∞,wn∈E⊥∞, then
yn
zn
→ asn→ ∞. (.)
By (F∞±) there existδ> andN∈Nsuch that
(zn),yn= ±
∇F(x,zn) –B∞(x)zn,yn
dx≥δ for alln≥N.
This implies that
(zn)zn ≥δ for alln≥N.
This is a contradiction with (.).
(ii) We apply Proposition .. SetL:=I–TB∞and
(z) =
B∞(x)z,z– F(x,z)dx.
Thencan be rewritten as
(z) =
Lz,z+(z), z∈E. (.)
By Lemma .,satisfies the Cerami condition and hence possesses the deformation property. From (F∞) one sees thatsatisfies
(z) =oz, z → ∞.
Now we show thatsatisfies the angle condition (AC–∞) at infinity with respect to the orthogonal decompositionE=E∞⊕E⊥∞when (F∞+) holds. Suppose it is not true, then for eachn∈N, there arezn∈E,zn=yn+wn,yn∈E∞,wn∈E⊥∞such that
zn ≥n, wn ≤
and
(zn),yn
> for alln∈N. (.)
It follows from (.) that
zn → ∞,
yn
zn→
, n→ ∞.
By (F+
∞) we have that forN∈N,
(zn),yn
=(zn),yn
= –
∇F(x,zn) –B∞(x)zn,yn
dx≤–δ, n≥N, this contradicts (.). Therefore (AC–
∞) holds, and by Proposition . we have
Cq(,∞) =δq,m∞+n∞Z, q∈Z.
(iii) This case is proved in a similar way.
The proof is finished.
4 Proofs of main theorems
In this section we give the proofs of main theorems in this paper.
Proof of Theorem. By (F∞), the functionaltakes the form
(z) =
Lz,z+(z), z∈E,
whereL=I–TB∞is a bounded self-adjoint linear operator,∈C(E,R) with a compact
gradientsatisfying
(z) =oz, z → ∞.
Sincen∞= , the problem is no resonance at infinity and thussatisfies the Palais-Smale condition. By Proposition .() we have
Cq(,∞)∼=δq,m∞Z, q∈Z. (.)
Thereforehas a critical pointz∗satisfying
Cm∞(,z∗). (.)
By Lemma . we have
Cq(, )∼= , ∀q∈Z. (.)
Proof of Theorem. By (F∞), (F∞) and Lemma ., we have
Cq(,∞)∼=δq,Z, q∈Z. (.)
Thereforehas a critical pointz∗satisfying
C(,z∗). (.)
We still have (.). Thusz∗= is a nontrivial weak solution of (GS). In fact,z∗is a global minimizer of.
Assume thatF(x,z) is even inz. We will employ Proposition . to prove the multiplicity in Theorem .. Nowis even,() = . By Lemma .,satisfies the Palais-Smale condition and is bounded from below following from the coercivity.
We verify (.). LetEkbe ak-dimensional subspace ofE. Forz∈Ek, as arguments in the
proof of Lemma ., we have
(z)≤ z
–c
zσLσ+czpLp. (.)
Sinceσ< <pand all norms onEkare equivalent, we get that forρk> small enough,
sup
z∈Xk∩Sρk
(z) < .
With all the conditions of Proposition . being verified, we get the conclusion thathas a sequence of critical valuesck< satisfyingck→ ask→ ∞. Thus (GS) has infinitely
many nontrivial weak solutions inE. The proof is finished.
Proof of Theorem. By a similar argument, it follows from Lemma . and Lemma ..
We conclude the paper with further comments and remarks.
Remark . (i) In Theorem ., whenλ(B∞) > which impliesm∞= andFis even inz, by the same arguments as the last part of the proof of Theorem ., one can show that (GS) has infinitely many nontrivial weak solutions inEwith negative energies which converge to zero.
(ii) In Theorem ., one nontrivial solution could be obtained if (F∞) is replaced by the nonquadraticity condition []
lim
|z|→∞
∇F(x,z),z– F(x,z)=∞ uniformly for a.e.x∈. (F˜∞+)
Indeed, (F˜∞+) is equivalent to
lim
|z|→∞
∇G(x,z),z– G(x,z)=∞ uniformly for a.e.x∈, (G˜∞)
which implies (F∞),i.e.,
(iii) The result for one nontrivial solution in Theorem . is valid whenFsatisfies (F), (F∞) and the nonquadraticity condition []
lim
|z|→∞
∇F(x,z),z– F(x,z)= –∞ uniformly for a.e.x∈. (F˜∞–)
Indeed, in this case,satisfies the Cerami condition andhas a saddle point structure at infinity with respect toE=E∞⊕E⊥∞in the sense thatis bounded from below onE⊥∞ and is anti-coercive onE∞. Then Proposition . in [] is applied to getC(,∞).
(iv) In Theorem ., the global condition (F∞±) is somewhat abstract and has been used in []. It could be verified if±(∇F(x,z) –B∞(x)z,z) acts as|z|s+near infinity for anys∈(, )
(see [, ]). See [] for more comparisons.
Remark . In Theorem ., we proved the multiplicity result by a critical point theorem in [] whenis even. This result is completely new for gradient systems. Since the critical groups ofat both zero and infinity are clearly computed, whenis even, the Morse equality may provide us an idea to give a different proofprovidedwe have in hand the following basic conclusion.
() Ifz∗is a solution of (HS), thenCq(,z∗)= for finitely manyq∈N.
Let () hold and letF be even. We prove the multiplicity for (GS) in Theorem . via Morse theory. Assume that (GS) has only finitely many pairs of nontrivial solutions. De-noteK={,±z,±z, . . . ,±zk}. Then by the Morse equality, one has that
∞
q=
(–)q
z∈K
dimCq(,z) = ∞
q=
(–)qdimCq(,∞).
By (), (.) and (.), it follows that
∞
q= (–)q
k
i=
dimCq(,zi) = ,
a contradiction. Similarly, if () is valid andFis even, then we have the same multiplicity result in Theorems . and ..
We note here that the conclusion () is valid foris ofC. A natural problem arises here whether or not that () is valid for aC functional. It is still open to the best of our knowledge. We will focus on this problem in near future.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
Supported by NSFC11271264, NSFC11171204 and PHR201106118.
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10.1186/1687-2770-2013-280
Cite this article as:Cai and Su:Gradient systems with sublinear term near the origin and asymptotically linear term
