R E S E A R C H
Open Access
Superlinear gradient system with a parameter
Anran Li
*and Jiabao Su
*Correspondence: [email protected]
School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China
Abstract
In this paper, we study the multiplicity of nontrivial solutions for a superlinear gradient system with saddle structure at the origin. We make use of a combination of bifurcation theory, topological linking and Morse theory.
MSC: 35J10; 35J65; 58E05
Keywords: gradient system; superlinear; critical group; Morse theory; linking
1 Introduction
In this paper, we study the existence of multiple solutions to the gradient system ⎧
⎨ ⎩
–uv=λA(x)uv+
Fu(x,u,v) Fv(x,u,v)
, x∈,
u=v= , x∈∂,
(GS)λ
where⊂RN is a bounded open domain with a smooth boundary∂andN≥,λis a real parameter andA∈M() is fixed.M() is 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}> .
We impose the following assumptions on the functionF:
(F) F∈C(×R,R).
(F) F(x, ) = ,∇F(x, ) = ,Fz(x, ) = forx∈.
(F) There isC> and <p<NN–:= *such that
∇F(x,z)≤C +|z|p–, forx∈,z= (u,v)∈R.
(F) There isμ> ,M> such that
<μF(x,z)≤∇F(x,z),z
for allx∈,|z| ≥M.
(F) Fz(x,z) > for|z|> small andx∈.
(F) Fz(x,z) < for|z|> small andx∈.
Here and in the sequel, is used to denote the origin in various spaces,| · |and (·,·) denote the norm and the inner product inR,Bzdenotes the matrix product inR for a ×
matrixBandz= (u,v)∈R. For two symmetric matricesBandCinR,B>Cmeans that B–Cis positive definite.
LetEbe the Hilbert spaceH
()×H() 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→Lp()×Lp() forp∈[, *), under the
assump-tions (F) and (F), the functional
(z) =
|∇z|–λA(x)z,zdx–
F(x,z)dx, z= (u,v)∈E (.)
is well defined and is of classC(see []) with derivatives
(z),w=
∇z∇w–λA(x)z,wdx–
∇F(x,z),wdx,
(z)z,z
=
∇z∇z–λ
A(x)z,z
dx–
Fz(x,z)z,z
dx,
forz= (u,v),w= (φ,ψ),z= (u,v),z= (u,v)∈E. Therefore, the solutions to (GS)λare
exactly critical points of inE.
By (F) the system (GS)λadmits a trivial solutionz= for any fixed parameterλ∈R. We
are interested in finding nontrivial solutions to (GS)λ. The existence of nontrivial solutions of (GS)λdepends on the behaviors ofFnear zero and infinity. The purpose of this paper is to find multiple nontrivial solutions to (GS)λwith superlinear term when the trivial solutionz= acts as a local saddle point of the energy functional in the sense that the parameterλis close toa higher eigenvalueof the linear gradient system with the given weight matrixA
⎧ ⎨ ⎩
–vu=λA(x)uv, x∈,
u=v= , x∈∂. (LA)
It is known (see [, ]) that for a given matrixA∈M(), (LA) admits a sequence of
distinct eigenvalues of finite multiplicity
<λA <λA <· · ·<λAk <· · ·
Denote byF–the negative part ofF,i.e.,F–(x,z) =max{–F(x,z), }.
We will prove the following theorems.
Theorem . Assume(F)-(F), (F)and let k≥be fixed.Then there isδ> such that whensup(x,z)∈×RF–(x,z)≤δ,for allλ∈(λAk+–δ,λAk+), (GS)λhas at least three nontrivial solutions in E.
Theorem . Assume(F)-(F), (F)and let k≥be fixed.Then there isδ> such that whensup(x,z)∈×RF–(x,z)≤δ,for allλ∈(λAk+,λAk++δ), (GS)λhas at least three nontrivial solutions in E.
Theorem . Assume(F)-(F)and F≤for x∈,z∈R with|z|small.Then there is
δ> such that whensup(x,z)∈×RF–(x,z)≤δ,for allλ∈(λAk+–δ,λAk+], (GS)λhas at least two nontrivial solutions in E.
We give some comments and comparisons. The superlinear problems have been stud-ied extensivelyviavariational methods since the pioneering work of Ambrosetti and Ra-binowitz []. Most known results on elliptic superlinear problems are contributed to a single equation with Dirichlet boundary data. Let us mention some historical progress on a single equation. When the trivial solution acted as a local minimizer of the energy functional, one positive solution and one negative solution were obtained by using the mountain-pass theorem in [] and the cut-off techniques; and a third solution was con-structed in a famous paper of Wang [] by using a two dimensional linking method and a Morse theoretic approach. When the trivial solution acted as a local saddle point of the energy functional, the existence of one nontrivial solution was obtained by applying a crit-ical point theorem, which is now well known as the generalized mountain-pass theorem, built by Rabinowitz in [] under a global sign condition (see []). Some extensions were done in [, ]vialocal linking. More recently, in the work of Rabinowitz, Su and Wang [], multiple solutions have been obtained by combining bifurcation methods, Morse theory and homological linking when is a saddle point in the sense that the parameterλis very close to a higher eigenvalue of the related linear operator.
In the current paper, we build multiplicity results for superlinear gradient systems by applying the ideas constructed in []. These results are new since, to the best of our knowl-edge, no multiplicity results for gradient systems have appeared in the literature for the case thatz= is a saddle point of .
We give some explanations regarding the conditions and conclusions. The assumptions (F)-(F) are standard in the study of superlinear problems. (F) and (F) are used for
bi-furcation analysis. It sees that (F) implies thatFis positive near zero, while (F) implies
thatFmust be negative near zero. The local properties ofFnear zero are necessary for constructing homological linking. WhenF≥, for any parameterλin a bounded inter-val, say in [λAk,λAk+), one can use the same arguments as in [] to construct linking starting fromλAk+. In our theorems, we do not require the global sign conditionF≥. When the parameterλis close to the eigenvalueλAk+, the homological linking will be constructed starting fromλAk+ and this linking is different from the one in []. This reveals the fact that whenλis close toλAk+from the right-hand side, the linking starting fromλAk+can still be constructed even ifFis negative somewhere. The conditions similar to (F) and
Hamiltonian systems were studiedviathe ideas in []. Since we treat a different problem in the current paper, we need to present the detailed discussions although some arguments may be similar to those in [, ].
The paper is organized as follows. In Section , we collect some basic abstract tools. In Section , we get solutions by linking arguments and give partial estimates of homological information. In Section , we get solutions by bifurcation theorem and give the estimates of the Morse index. The final proofs of Theorems .-. are given in Section .
2 Preliminary
In this section, we give some preliminaries that will be used to prove the main results of the paper. We first collect some basic results on the Morse theory for aCfunctional defined
on a Hilbert space.
Let E be a Hilbert space and ∈C(E,R). Denote K={z∈E| (z) = }, c ={z∈ E| (z)≤c},Kc={z∈K| (z) =c}forc∈R. We say that satisfies the (PS)ccondition
at the levelc∈Rif any sequence{zn} ⊂Esatisfying (zn)→c, (zn)→ asn→ ∞, has a convergent subsequence. satisfies (PS) if satisfies (PS)cat anyc∈R.
We assume that satisfies (PS) and#K<∞. Letz∈Kwith (z) =c∈RandUbe a
neighborhood ofzsuch thatU∩K={z}. The group
Cq( ,z) :=Hq
c
∩U, c∩U\ {z}
, q∈Z
is called theqth critical group of atz, whereH∗(A,B) denotes a singular relative ho-mology group of the pair (A,B) with coefficients fieldF(see [, ]).
Leta<inf (K). The group
Cq( ,∞) :=HqE, a, q∈Z
is called theqth critical group of at infinity (see []).
We callMq:=z∈KdimCq( ,z) theqth Morse-type numbers of the pair (E, a) and
βq:=dimCq( ,∞) the Betti numbers of the pair (E, a). The core of the Morse theory
[, ] is the following relations betweenMqandβq:
(Morse inequality)
q
j=
(–)q–jMj≥ q
j=
(–)q–jβj, forq∈Z;
(Morse equality) ∞
q=
(–)qMq= ∞
q=
(–)qβq.
If K=∅, thenβq= for allq. SinceMq≥βq for eachq∈Z, it follows that ifβq∗= for someq∗∈Z, then must have a critical pointz∗withCq∗( ,z∗)∼= . IfK={z∗}, then
Cq( ,∞)∼=Cq( ,z∗) for allq. Thus, ifCq( ,∞)∼=Cq( ,z∗) for someq, then must have a new critical point. One can use critical groups to distinguish critical points obtained by other methods and use the Morse equality to find new critical points.
Proposition . Assume that z is an isolated critical point of ∈C(E,R)with a finite Morse index m(z)and nullity n(z).Then
() Cq( ,z)∼=δq,m(z)Fifn(z) = ;
() Cq( ,z)∼= forq∈/[m(z),m(z) +n(z)](Gromoll-Meyer []);
() ifCm(z)( ,z)∼= thenCq( ,z)∼=δq,m(z)F;
() ifCm(z)+n(z)( ,z)∼= thenCq( ,z)∼=δq,m(z)+n(z)F.
Proposition .([, ]) Letbe an isolated critical point of ∈C(E,R)with a finite Morse index mand nullity n.Assume that has a local linking atwith respect to a direct sum decomposition E=E–⊕E+,κ=dimE–,i.e.,there exists r> small such that
(z) > for z∈E+, <z ≤r, (z)≤ for z∈E–,z ≤r.
Then Cq( , )∼=δq,κZfor eitherκ=morκ=m+n.
The concept of local linking was introduced in []. In [] a partial result was given for aCfunctional. The above result was obtained in [].
Now, we recall an abstract linking theorem which is from [, , ].
Proposition .([, , ]) Let E be a real Banach space with E=X⊕Y and=dimX be finite.Suppose that ∈C(E,R)satisfies(PS)and
( ) there existρ> andα> such that
(z)≥α, z∈Sρ=Y∩∂Bρ, (.)
whereBρ={z∈E|z ≤ρ},
( ) there existR>ρ> ande∈Ywithe= such that
(z) <α, z∈∂Q, (.)
where
Q=z=y+se|z ≤R,y∈X, ≤s≤R.
Then has a critical point z∗with (z∗) =c∗≥αand
C+( ,z∗)= . (.)
We note here that under the framework of Proposition .,Sρand∂Q homotopically link with respect to the direct sum decompositionE=X⊕Y.Sρand∂Qare alsohomologically
linked. The conclusion (.) follows from Theorems .and . of Chapter II in []. (See also [].)
We finally collect some properties of the eigenvalue problem (LA). Associated with a
matrixA∈M(), there is a compact self-adjoint operatorTA:E→Esuch that
TAz,w =
The compactness ofTAfollows from the compact embeddingE→L()×L(). The
operatorTApossesses the property thatλAis an eigenvalue of (L
A) if and only if there is
nonzeroz∈Esuch that
λATAz=z.
(LA) has the sequence of distinct eigenvalues
<λA <λA <· · ·<λAk <· · · → ∞,
and each eigenvalueλAof (L
A) has a finite multiplicity. Forj∈N, denote
EλAj=ker––λAjA, Ej=
j
i=
EλAi, j=dimEj.
Set
Qλ(z) =
|∇z|–λA(x)z,zdx, z∈E.
Then the following variational inequalities hold:
Qλ(z)≤λ
A j –λ
λA j
z, z∈Ej,
Qλ(z)≥λ
A j+–λ
λAj+ z
, z∈E⊥
j .
We refer to [, ] for more properties related to the eigenvalue problem (LA) and the
op-eratorTA.
3 Solutionsviahomological linking
In this section, we give the existence a nontrivial solution of (GS)λby applying homological linking arguments and then give some estimate of its Morse index. The following lemmas are needed.
Lemma . Assume that F satisfies(F)-(F),then for any fixedλ∈R,the functional satisfies the(PS)condition.
Proof By (F) and the compact embeddingE→Lp()×Lp() for ≤p< *, it is enough
to show that any sequence{zn} ⊂Ewith
(zn)≤C, n∈N, (zn)→, n→ ∞ (.)
is bounded inE. Here and below, we useCto denote various positive constants. We modify the arguments in []. Choosing a positive numberβ∈(/μ, /) fornlarge, we have that
By (F) we deduce that
F(x,z)≥C|z|μ–C, forz∈R,x∈. (.)
Therefore,
C+βzn
≥ (zn) –β (zn),zn
=
–β
|∇zn|–λA(x)zn,zndx–
F(x,zn) –β∇F(x,zn),zndx
≥
–β
zn–
–β
|λ|zn+C(μβ– )znμμ–C,
where=maxx∈ ¯A(x)R. By the Hölder inequality and the Young inequality, we get for
any> that
zn≤ ||μ
– μ zn
μ≤
||(μ– )
μ
–μ +
μzn μ
μ. (.)
Thus, for a fixed> small enough, we have by (.) that
C+βzn ≥
–β
zn+
C(μβ– )zn μ μ.
Therefore,{zn}is bounded inE. The proof is complete.
Now, we construct a homological linking with respect to the direct sum decomposition ofEfork≥:
E=Ek+⊕Ek⊥+, dimEk+=k+.
Take an eigenvectorφk+corresponding to the eigenvalueλAk+of (LA) withφk+= . Set
Vk+:=Ek+⊕span{φk+}.
Lemma . Assume that F satisfies(F)-(F)and k≥.Then there exist constantsρ>
small,α> such that for allλ≤λ∗:=λ
A k++λAk+
,such that
(z)≥α, for z∈Ek⊥+withz=ρ. (.)
Proof By the conditions (F) and (F), for> , there isC> such that
F(x,z)≤ S
|z|+C|z|p,
whereS is the constant for the embeddingE→L()×L() such that|z|≤Sz
forz∈E. Since forz∈Ek⊥+,
Qλ(z) =
|∇z|–λA(x)z,zdx≥λ A k+–λ
λAk+ z
it follows that
(z) = Qλ(z) –
F(x,z)dx
≥
λA k+–λ
λA k+
z– S
|z|dx–C
|z|pdx
≥
λAk+–λ– λAk+ z
–Czp
=
η(λ,)z
–C
zp,
whereCis independent ofλand
η(λ,) =λ
A
k+–λ–
λAk+ . (.)
Sincep> and the functiong(r) = η(λ,)r
–C
rpachieves its maximum
gmax=p– p (pC)
–pη(λ,)
p
p–:=α(λ,) (.)
on (,∞) at
ρλ,=
η(λ,)
pC
p–
, (.)
we see that
(z)≥α(λ,), forz∈Ek⊥+withz=ρλ,. (.)
Sinceη(λ,) is a decreasing function with respect toλfor any fixed> small, (.) holds for
α:=α(λ∗,∗), ρ:=ρλ∗,∗, here∗=
λAk+–λAk+.
The constantsαandρare independent ofλ≤λ∗. The proof is complete.
Lemma . Assume that F satisfies(F), (F)and k≥.Then there exist R> ,δ> and
σ∈R,all independent ofλ,such that whenλ∈(λAk+–δ,λkA++δ)andsup(x,z)∈×RF–(x, z)≤δ,
(z)≤σ<α, for z∈∂Q, (.)
where
Q=z∈Vk+|z ≤R,z=y+sφk+,y∈Ek+,s≥
Proof From (F) we deduce (.) with a positive constantCindependent ofλ. Forz∈Vk+,
writez=y+w+φ,y∈Ek,w∈E(λAk+),φ∈span{φk+}. Assume thatλ∈(λAk,λAk+), then
(z) = Qλ(y) +
Qλ(w) +
Qλ(φ) –
F(x,z)dx
≤ λAk–λ
λAk y
+λAk+–λ
λAk+ w
+λAk+–λ
λAk+ φ
–Czμ μ+C
≤ λAk+–λAk
λAk+ z
–Czμ
μ+C. (.)
Sinceμ> anddimVk+<∞, (.) shows that there existsR> independent ofλsuch
that
(z)≤, forz∈Vk+withz=R. (.)
Now, fix such anR> thatR>ρwithρgiven in Lemma .. Fory∈Ek+withy ≤R,
we writey=w+φ,w∈Ek,φ∈E(λAk+). Set:=sup(x,z)∈×RF–(x,z). Then we have that
(y) = Qλ(w) +
Qλ(φ) –
F(x,y)dx
≤λAk –λ
λAk w
+λAk+–λ
λAk+ φ
+
{x∈:F≤}F
–(x,y)dx
≤|λAk+–λ|
λA k+
R+||. (.)
Since∂Q={z=y+sφk+|z=R,y∈Ek,s≥} ∪ {y∈Ek|y ≤R}, taking
δ=
R
λA k+
+ || –
α, σ=α ,
then, whenλ∈(λAk+–δ,λAk++δ) and≤δ,
(z)≤σ<α, forz∈∂Q.
The proof is complete.
Now, we apply Proposition . to get the following existence result with partial homo-logical information.
Theorem . Let F satisfy(F)-(F)and k≥.Then there isδ> such that when≤δ, for eachλ∈(λAk+–δ,λAk++δ), (GS)λhas one nontrivial solution z*with a critical group satisfying
Ck++
Proof By Lemma ., satisfies (PS). By Lemmas . and ., for each fixedλ∈(λk+–
δ,λk++δ), satisfies ( ) and ( ) in the sense that
⎧ ⎨ ⎩
( ) : (z)≥α, z∈Sk+:=Sρ∩Ek+;
( ) : (y)≤σ<α, y∈∂Q.
(.)
SinceSk+and∂Qhomotopically link with respect to the decompositionE=Ek+⊕E⊥k+,
anddimVk+=k++ , it follows from Proposition . that has a critical pointz*∈E
with positive energy (z*)≥α> and its critical group satisfying (.). The proof is
complete.
We give some remarks. The existence of one nontrivial solution in Theorem . is valid whenF is of classC. From Lemma ., one sees that the energy of the obtained
solu-tion is bounded away from asλis close toλAk+. A rough local sign condition onFis needed. IfF≥, then for any fixedλ∈[λAk,λAk+), a linking with respect toEk⊕E⊥k can be
constructed. Proposition . is applied again to get a nontrivial solutionz∗satisfying
Ck+( ,z∗)∼= . (.)
Therefore, when a global sign conditionF≥ is present, asλis close toλA
k+from the
left-hand side, two linkings can be constructed and two nontrivial solutions can be obtained. The question is how to distinguishz*fromz
∗. Theorem . includes the case that forλ close toλAk+from the right-hand side, the linking with respect toEk+⊕E⊥k+is constructed
provided the negative values ofFare small. This phenomenon was first observed in [].
4 Solutionsviabifurcation
In this section, we get two solutions for (GS)λviabifurcation arguments []. We first cite the bifurcation theorem in [].
Proposition .(Theorem . in []) Let E be a Hilbert space and∈C(E,R)with
∇(u) =Lu+H(u),
where L∈L(E,E)is symmetric and H(u) =o(u)asu →.Consider the equation
Lu+H(u) =λu. (.)
Letμ∈σ(L)be an isolated eigenvalue of finite multiplicity.Then either
(i) (μ, )is not an isolated solution of(.)in{μ} ×E,or
(ii) there is a one-sided neighborhoodofμsuch that for allλ∈\ {μ}, (.)has at least two distinct nontrivial solutions,or
(iii) there is a neighborhoodofμsuch that for allλ∈\ {μ}, (.)has at least one nontrivial solution.
Theorem . Assume that F satisfies(F)-(F).Let k≥be fixed.Then there existsδ> such that(GS)λhas at least two nontrivial solutions for
() everyλ∈(λAk+–δ,λAk+)if(F)holds;
() everyλ∈(λAk+,λkA++δ)if(F)holds.
Furthermore,the Morse index m(zλ)and the nullity n(zλ)of such a solution zλsatisfy
k≤m(zλ)≤m(zλ) +n(zλ)≤k+, for <λ–λAk+<δ. (.)
Proof Under the assumptions (F)-(F), for each eigenvalueλAj of (LA), (λAj, ) is a
bifur-cation point of (GS)λ(see []).
Let (λ,z)∈R×Ebe asolutionof (GS)λnear (λAk+, ) which satisfies ⎧
⎨ ⎩
–z=λA(x)z+∇F(x,z), x∈,
z= (u,v) = , x∈∂. (.)
By (F) and (F), we have
∇F(x,z) =∇F(x, ) +Fz(t,ηz)x=Fz(x,ηz)z, for some <η< . (.)
Let (F) hold. By the elliptic regularity theory (see []),z> small implieszC>
small. Then by (F), we have that
Fzz,ηz(x)> , x∈. (.)
Now, consider the linear eigenvalue gradient system: ⎧
⎨ ⎩
–y–Fz(x,ηz(x))y=μA(x)y, x∈,
y= , x∈∂. (.)
We denote the distinct eigenvalues of (.) byμ(z) <μ(z) <· · ·<μi(z) <· · · asz= .
By (F), if we takez= , then for eachi∈N, there isj∈Nsuch thatμi() =λAj. By (.),
the standard variational characterization of the eigenvalues of (.) shows thatμi(z) is less thanthe correspondingjth ordered eigenvalueλA
j of (LA). Furthermore,μi(z)→λAj
asz→ inE. By (.) and (.), we see thatzis a solution of (.) with eigenvalueλ. It must be thatλ<λAk+ sinceλis close toλAk+. Therefore, the case (ii) of Proposition . occurs under the given conditions. This proves the case (). The existence for the case () is proved in the same way.
Now, we estimate the Morse indices for the solutions obtained above. Letzλbe a bifur-cation solution of (GS)λ. Then
zλ → asλ→λAk+.
Applying the elliptic regularity theory, we have that
For eachy∈E, we have
(zλ)y,y=Qλ(y) –
Fz(x,zλ)y,ydx.
Therefore, fory∈Ek,
(zλ)y,y≤λ A k –λ
λAk y
+
Fz(x,zλ)y,ydx,
and forφ∈E⊥k+,
(zλ)φ,φ≥λ
A k+–λ
λA k+
φ–
Fz(x,zλ)φ,φdx.
By (F) and (.), there existsδ> such that when <|λ–λAk+|<δ,
(zλ)y,y< , ∀y∈Ek\ {},
(zλ)φ,φ> , ∀φ∈E⊥k+\ {}.
Therefore, the Morse index m(zλ) and the nullityn(zλ) ofzλsatisfy (.). The proof is
complete.
5 Proofs of main theorems
In this section, we give the proof of main theorems in this paper. We first compute the critical groups of at both infinity and zero.
Lemma .(see []) Let F satisfy(F)-(F),then for any fixedλ∈R,
Cq( ,∞)∼= , for all q∈Z. (.)
Proof The idea of the proof comes from the famous paper []. We include a sketched proof in an abstract version. Givenλ∈R, denoteB={z∈E:z ≤},S=∂B. Modifying the
arguments in [], we get the following facts:
(sz)→–∞ ass→+∞,∀z∈S. (.)
Fora–, (sz)≤a =⇒ d
ds (sz) < . (.)
The following arguments are from []. As () = , it follows from (.) and (.) that for eachz∈S, there is auniqueτ(z) > such that
τ(z)z=a. (.)
By (.) and the implicit function theorem, we have thatτ∈C(S,R). Define
π(z) = ⎧ ⎨ ⎩
, if (z)≤a,
Thenπ∈C(E\ {},R). Define a map: [, ]×E\ {} →E\ {}by
(t,z) = ( –t)z+tπ(z)z. (.)
Clearly,is continuous, and for allz∈E\ {}with (z) >a, by (.),
(,z)= πz–zz–z=a.
Therefore,
(,z)∈ a for allz∈E\ {}, (t,z) =z for allt∈[, ],z∈ a,
and so ais a strong deformation retract ofE\ {}. Hence,
Cq( ,∞) :=HqE, a∼=HqE,E\ {}∼=Hq(B,S)∼= , q∈Z
sinceSis contractible, which follows from the fact thatdimE=∞.
Lemma . Let F satisfy(F)-(F).
() Forλ∈(λAk,λAk+),Cq( , )∼=δq,kF.
() Forλ∈(λAk+,λAk+),Cq( , )∼=δq,k+F.
() Forλ=λAk+,ifF(x,z)≤for|z|small,thenCq( , )∼=δq,kF.
() Forλ=λAk+,ifF(x,z)≥for|z|small,thenCq( , )∼=δq,k+F. Proof By (F), we have
()y,y=Qλ(y) =
|∇y|–λA(x)y,ydx, y∈E.
() Whenλ∈(λAk,λAk+),z= is a nondegenerate critical point of with the Morse index
m=k, thusCq( , )∼=δq,kF.
() Whenλ∈(λA
k+,λAk+),z= is a nondegenerate critical point of with the Morse
indexm=k+, thusCq( , )∼=δq,k+F.
() Whenλ=λA
k+,z= is a degenerate critical point of with the Morse indexm=k
and the nullityn=dimE(λAk+),m+n=k+.
Assume that F(x,z)≤ for|z| ≤σ with σ > small. We will show that has a lo-cal linking structure atz= with respect toE=Ek⊕E⊥k. If this has been done, then by
Proposition ., we haveCq( , )∼=δq,kF. Now, can be written as
(z) =
QλAk+(z) –
F(x,z)dx.
By (F) and (F), for> , there isC> such that
F(x,z)≤ |z|
+C
Hence, forz∈Ek, we have that
(z)≤λ
A k–λAk+
λAk z
+
|z|
+C|z|pp.
SinceEkis finite dimensional, all norms onEkare equivalent, hence forr> small,
(z)≤, forz∈Ek,z ≤r.
By (F), we have that for someC> ,
F(x,z)≤C|z|p, |z| ≥σ,x∈. (.)
Forz∈E⊥k, we writez=y+wwherey∈E(λAk+) andw∈E⊥k+. Then
(z)≥λ
A k+–λAk+
λAk+ w
–
F(x,z)dx. (.)
Forx∈with|z(x)| ≥σ, we have|w(x)| ≥|z(x)|. Hence, by (.) and the Poincaré
in-equality, we have for various constantsC> ,
{x:|z(x)|≥σ}
F(x,z)dx
≤C
{x:|z(x)|≥σ}
z(x)pdx
≤C
w(x)pdx
≤Cwp. (.)
Forx∈with|z(x)| ≤σ,F(x,z(x))≤. Therefore,
(z)≥ λ
A k+–λAk+
λAk+ w
–
{x:|z(x)|≤σ}
Fx,z(x)dx–Cwp
≥ λAk+–λAk+
λAk+ w
–Cwp. (.)
Sincep> , forr> small,
(z) > , forz=y+wwithw= ,z ≤r. (.)
Forz=y∈E(λAk+), it must hold that
(y) = –
Fx,y(x)dx> if <y ≤r. (.)
x∈,∇F(x,y)≡ for allx∈. Thus, would not be an isolated critical point of and (GS)λwould have infinitely many nontrivial solutions. By (.) and (.), we verify that
(z) > , forz∈Ek⊥, <z ≤r.
Applying Proposition ., we obtain
Cq( , ) =δq,kF.
() When F(x,z)≥ for|z| small, a similar argument shows that has a local link-ing structure atz= with respect toE=Ek+⊕E⊥k+. By Proposition ., it follows that
Cq( , )∼=δq,k+F.
Finally, we prove the theorems.
Proof of Theorem. It follows from (F) thatF(x,z)≥ for|z|> small. By Theorem .
for the partλ∈(λAk+–δ,λAk+), (GS)λhas a nontrivial solutionzλsatisfying
Ck++
,zλ∼= . (.)
By Theorem .(), (GS)λhas two nontrivial solutionsziλ(i= , ) with their Morse indices satisfying
k≤m
zi
λ
≤mzi
λ
+nzi
λ
≤k+, i= , .
From Proposition .(), we have that
Cq ,ziλ ∼
= , q∈/[k,k+],i= , . (.)
From (.) and (.), we see thatzλ=zi
λ(i= , ). The proof is complete.
Proof of Theorem. With the same argument as above, it follows from Theorem .() and Theorem . for the partλ∈(λA
k+,λAk++δ). We omit the details.
Proof of Theorem. By Theorem . for the partλ∈(λAk+–δ,λAk+], (GS)λhas a solution
zλwith its energy (zλ)≥α> and
Ck++
,zλ∼
= . (.)
By Lemma . and Lemma .(), we have that
Cq( ,∞)∼= , forq∈Z, (.)
Cq( , )∼=δq,kF, forq∈Z. (.)
we have ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
Cq( ,∞)∼=Hq(E, a);
Cq( , )∼=Hq( b, a);
Cq( ,zλ)∼=Hq(E, b).
(.)
By (.), the long exact sequences for the topological triple (E, b, a) read as
· · · →Cq+( ,∞)→Cq+
,zλ→Cq( , )→Cq( ,∞)→ · · ·. (.)
We deduce by (.) and (.) that
Cq+
,zλ∼=Cq( , ), forq∈Z. (.)
Takeq=k+in (.), then
Ck++
,zλ∼
=Ck+( , )∼=
which contradicts (.). The proof is complete.
We finally remark that Theorem . is valid forλ∈(λA
–δ,λA), from which one sees that z= is a local minimizer of .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to the anonymous referee for his/her valuable suggestions. The second author was supported by NSFC11271264, NSFC11171204, KZ201010028027 and PHR201106118.
Received: 30 July 2012 Accepted: 25 September 2012 Published: 9 October 2012
References
1. Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math., vol. 65. Am. Math. Soc., Providence (1986)
2. Chang, KC: An extension of the Hess-Kato theorem to elliptic systems and its applications to multiple solutions problems. Acta Math. Sin.15, 439-454 (1999)
3. Chang, KC: Principal eigenvalue for weight in elliptic systems. Nonlinear Anal.46, 419-433 (2001)
4. Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal.14, 349-381 (1973)
5. Wang, ZQ: On a superlinear elliptic equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire8, 43-57 (1991) 6. Rabinowitz, PH: Some critical point theorems and applications to semilinear elliptic partial differential equations.
Ann. Sc. Norm. Super. Pisa, Cl. Sci.5, 215-223 (1978)
7. Li, SJ, Liu, JQ: Some existence theorems on multiple critical points and their applications. Kexue Tongbao17, 1025-1027 (1984)
8. Li, SJ, Willem, M: Applications of local linking to critical point theory. J. Math. Anal. Appl.189, 6-32 (1995) 9. Rabinowitz, PH, Su, JB, Wang, ZQ: Multiple solutions of superlinear elliptic equations. Rend. Lincei Mat. Appl.18,
97-108 (2007)
10. Li, XL, Su, JB, Tian, RS: Multiple periodic solutions of the second order Hamiltonian systems with superlinear terms. J. Math. Anal. Appl.385, 1-11 (2012)
11. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems, 1st edn. Springer, Berlin (1989)
12. Chang, KC: Infinite Dimensional Morse Theory and Multiple Solutions Problems, 1st edn. Birkhäuser, Boston (1993) 13. Bartsch, T, Li, SJ: Critical point theory for asymptotically quadratic functionals and applications to problems with
14. Gromoll, D, Meyer, M: On differential functions with isolated point. Topology8, 361-369 (1969) 15. Liu, JQ: A Morse index for a saddle point. Syst. Sci. Math. Sci.2, 32-39 (1989)
16. Su, JB: Multiplicity results for asymptotically linear elliptic problems at resonance. J. Math. Anal. Appl.278, 397-408 (2003)
17. Gilbarg, D, Trudinger, NS: Elliptic Partial Differential Equations of the Second Order, 2nd edn. Springer, Berlin (1983)
doi:10.1186/1687-2770-2012-110