R E S E A R C H
Open Access
Homoclinic solutions for second order
Hamiltonian systems near the origin
Li-Li Wan
**Correspondence: [email protected] School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China
Abstract
Under some local superquadratic conditions onW(t,u) with respect tou, the existence of infinitely many homoclinic solutions is obtained for the nonperiodic second order Hamiltonian systemsu¨(t) –L(t)u(t) +∇W(t,u(t)) = 0,∀t∈R, whereL(t) is unnecessarily coercive.
Keywords: homoclinic solutions; second order Hamiltonian systems; local conditions
1 Introduction and main results
Let us consider the second order Hamiltonian systems
¨
u(t) –L(t)u+∇Wt,u(t)= , ∀t∈R, ()
whereL∈C(R,RN
) is a symmetric matrix-valued function and∇W(t,x) = ∂
∂xW(t,x). As usual, we say thatuis a nontrivial homoclinic solution (to ) ifu∈C(R,RN),u= , u(t)→ as|t| → ∞. In the following, (·,·) :RN×RN →Rdenotes the standard inner product inRN and| · |is the induced norm, and letIN be the identity matrix of orderN.
With the variational methods, the existence and multiplicity of homoclinic orbits of problem () have been obtained in many papers (see [–]), mainly in the case thatW satisfies some global assumptions for alltandu. Among other results, under some local conditions onW, Lv and Jiang [] investigated the existence of one nontrivial homoclinic solution for the second order Hamiltonian systems
¨
u–L(t)u(t) +∇Wt,u(t)=f(t), ∀t∈R
as a limit of periodic solutions of a certain sequence of boundary-value problems. Later, it was proved in [] that ifL(t) is coercive andW(t,u) is subquadratic near the origin with respect tou, then problem () has a sequence of homoclinic solutions converging to zero inL∞ norm. There were no conditions assumed onW forularge. More precisely, one presented the following assumptions:
(A) There exists a constantα< such thatl(t)|t|α–→ ∞as|t| → ∞, where
l(t) = inf x∈RN,|x|=
L(t)x,x.
(A) There are constantsc> and ≤ν∈(–α, )such that
∇W(t,x)≤c|x|ν, ∀(t,x)∈R×Bδ(),
whereBδ()denotes the ball inRN centered at with radiusδ> .
(A)
lim
|x|→
W(t,x)
|x| =∞ uniformly fort∈R.
(A) W(t,x) – (∇W(t,x),x) > for allt∈Randx∈RN\ {}.
In this note, we will consider problem () where L(t) is unnecessarily coercive, and W(t,u) is superquadratic near the origin. The exact assumptions onLandWare as fol-lows.
Theorem Assume the following conditions hold:
(L) There existsl≥such that
l(t) := inf x∈RN,|x|=
L(t)x,x≥–l, ∀t∈R.
(L) There exists a constantξ> such that
meast∈R| |t|–ξL(t)MIN
< +∞, ∀M> .
(W) W ∈C(R×Bδ(),R)is even inuandW(t, ) = ,whereBδ()denotes the ball in
RNcentered atwith radiusδ> .
(W) There are constantsc> and <θ< such that
∇W(t,x)≤c|x|θ, ∀(t,x)∈R×Bδ().
(W) There exists a constantp> such that
lim
|x|→
W(t,x)
|x|p = uniformly fort∈R.
(W) W(t,x) – (∇W(t,x),x) < for allt∈Randx∈RN\ {}. (W) There exists a constantμ> such that
lim
|x|→
W(t,x)
|x|μ =∞ uniformly fort∈R.
Then problem()has a sequence of homoclinic solutions{uk}such thatmaxt∈R|uk(t)| →
as k→ ∞.
Theorem is different from Theorem . in []. As far as the authors know, there is little research concerning the multiplicity of homoclinic solutions for problem () simultane-ously under local conditions and noncoercive conditions, so our result is different from the previous results in the literature.
The proof is motivated by the argument in []. We will modify and extendWto an ap-propriateWand show for the associated modified functionalIthe existence of a sequence of homoclinic solutions converging to zero inL∞norm, and therefore we obtain infinitely many homoclinic solutions for the original problem.
2 Proof of theorems
First of all, we introduce the Sobolev space that we study. LetAbe the self-adjoint ex-tension of the operator –(dtd) +L(t) with the domainD(A)⊂L≡L(R,RN). Denote by
{E(λ)|–∞<λ<∞}and|A|the spectral resolution and the absolute value ofA, respec-tively. DefineU=I–E() –E(–).Ucommutes withA,|A|and|A|/, andA=U|A|is
the polar decomposition ofA(see []). LetE=D(|A|/), the domain of|A|/. Define on
Ethe inner product and the corresponding norm:
(u,v)=
|A|/u,|A|/v+ (u,v),
u= (u,u)/ ,
whereu,v∈Eand (·,·)denotes the inner product ofL. ThenEis a Hilbert space, and it
is easy to verify thatEis continuously embedded inH(R,RN).
Lemma (see []) Suppose that L(t)satisfies(L)and(L).Then E is compactly
embed-ded in Lq(R,RN)for q∈[,∞].
Remark By Lemma it is easy to prove that the spectrumσ(A) consists of eigenval-ues numbered by λ≤λ≤ · · · →+∞, and a corresponding system of eigenfunctions,
(ej)(Aej=λjej), forms an orthogonal basis inL. Letj–=#{j|λj< },j=#{j|λj= }and ¯
j=j–+j, where#Adenotes the number of elements of the setA. SetE–=span{e, . . . ,ej–},
E=span{ej–
+, . . . ,e¯j}=kerA, andE+=span{ej¯+, . . .}. Then one hasE=E–⊕E⊕E+. We
introduce onEthe following inner product and the corresponding norm:
(u,v) =|A|u,|A|v +
u,v, u= (u,u) =|A|u
+u
,
whereu=u–+u+u+andv=v–+v+v+∈E=E–⊕E⊕E+. Obviously the norms ·
and · are equivalent and so the norm · onEwill always be used. By Lemma we
see that there exists a constantγq> such that
uq≤γqu, ∀u∈E,∀q∈[,∞]. ()
Lemma Assume that(W)-(W)are satisfied.There is <r< δ andW ∈C(R,RN)
(i)
∇W(t,x)≤c
|x|θ+|x|p–, ∀(t,x)∈R×RN, ()
wherecis a constant; (ii)
W(t,x) := W(t, x) –∇W(t,x),x≤, ∀(t,x)∈R×RN ()
and
W(t,x) = iff |x|= . ()
Proof By the mean value theorem, (W) and (W) imply that
W(t,x)≤c|x|θ+, ∀(t,x)∈R×Bδ(). ()
Next we modifyW(t,x) forxoutside a neighborhood of the origin . Choose
<β< γpp
,
whereγpis the constant given in (). By (W), there is a constantr∈(,δ) such that
W(t,x)≤β|x|p, ∀t∈Rand|x| ≤r. ()
Define a cut-off functionρ∈C(R,R) satisfying
ρ(t) =
, ≤t≤r, , t≥r,
and –r≤ρ(t) < forr<t< r. Usingρ, we define
W(t,x) :=ρ|x|W(t,x) + –ρ|x|W∞(x), ∀(t,x)∈R×RN, ()
whereW∞(x) =β|x|p. Then by direct computation we get
∇W(t, x) =ρ|x|∇W(t,x) +ρ|x|W(t,x) + –ρ|x|W∞ (x) –ρ|x|W∞(x), ()
W(t,x) =ρ|x|W(t,x) –∇W(t,x),x+ ( –p) –ρ|x|W∞(x)
–ρ|x|W(t,x) –W∞(x)|x| () for (t,x)∈R×RN. It follows from (W
) and (W) that
∇W(t, ) = W(t, ) = , ∀t∈R. ()
Then by (), (), (W), and the choice of the cut-off functionρ, we have
and
∇W(t,x)≤∇W(t,x)+
rW(t,x)+W
∞(x) +
rW∞(x) ≤c|x|θ+ c|x|θ+βp|x|p–+ β|x|p–
= c|x|θ+ ( +p)β|x|p–, ∀t∈R,|x|< r.
Therefore, () is satisfied ifc=max{c, ( +p)β}.
Finally, we prove () and (). On one hand, using () we know thatW(t,x) = whenever x= . On the other hand, assume thatr<|x|< r. By (), (W), (), and the choice of the
cut-off functionρ, we obtain
ρ|x|W(t,x) –∇W(t,x),x< , ( –p) –ρ|x|W∞(x)≤,
and
–ρ|x|W(t,x) –W∞(x)|x| ≤.
The above estimates imply thatW(t,x) < ifr<|x|< r. Besides, when|x| ≥r, by () we have
W(t,x) = ( –p)W∞(x) < .
When <|x| ≤r, by (W) we get
W(t,x) = W(t,x) –∇W(t,x),x< .
Thus () and () are verified. The proof is completed.
We now consider the modified problem
¨
u(t) –L(t)u+∇Wt,u(t)= , ∀t∈R, ()
whose solutions correspond to critical points of the functional
I(u) = R
|˙u|+L(t)u,udt–
R
W(t,u)dt
= u
+
– u
–
–
R
W(t,u)dt
for allu=u–+u+u+∈E=E–+E+E+. By () and () we have
W(t,u)≤c|u|θ++β|u|p, ∀(t,u)∈R×RN. ()
RewriteIas follows:
In the following,cwill be used to denote various positive constants where the exact values are different. are homoclinic solutions of problem().
Proof By (), for anyη∈[, ],u,h∈RN we have
∇W(t,u+ηh),h≤c|u|θ|h|+|h|θ++|u|p–|h|+|h|p,
where cis independent of η. Hence, for any u,h∈E, by the mean value theorem and Lebesgue’s dominated convergence theorem, we get
Bis bounded and continuous fromLp(R) toL
p
p–(R). For anyv,h∈E,
dI(u) –dI(v),h=
R(Bu–Bv,h)dt
=
R(Bu+Bu–Bv–Bv,h)dt
≤
R|Bu–Bv||h|dt+ R|Bu–Bv||h|dt
≤cBu–Bvθ+
θ h+cBu–Bv
p p–h,
which implies that
dI(u) –dI(v)E∗≤cBu–Bvθ+
θ +cBu–Bv
p p–.
Now supposeunuinE, then by Lemma ,un→uinL(θ+)(R) andLp(R). Combining the above arguments, we see thatdI(u) is weakly continuous. Therefore,I(u)∈C(E,R)
andI :E→E∗is compact.
Finally, we show that nontrivial critical points of I onE are homoclinic solutions of problem (). Letu∈Ebe a nontrivial critical point ofI. A standard argument shows that u∈C(R,RN) and satisfies problem () (see [] for more details). SinceEis continuously embedded inH(R,RN),u(t)→ as|t| → ∞. The proof is completed.
Lemma Assume that(L), (L), (W)-(W)are satisfied.Thenis the only critical point
of I such that I(u) = .
Proof By (W), (W), and Lemma , we know that is a critical point ofIwithI() = .
Now letu∈Ebe a critical point ofIwithI(u) = . Then we have
= I(u) –I(u),u= –
R
W(t,u)dt,
whereW is defined in (). This together with (ii) of Lemma implies that|u(t)|= for all
t∈R. The proof is completed.
The following lemma is due to Bartsch and Willem [] and we quote it from [].
Lemma Let E be a Banach space with a finite-dimensional approximation in the sense that E=j∈NE(j),where E(j)are all finite-dimensional subspaces.Let I∈C(E,R)be an
even functional and satisfy:
(F) For everyk≥k,there existsRk> such thatI(u)≥for everyu∈Ek:=
j≥kE(j)
withu=Rk,andbk:=infu∈BkI(u)→ask→ ∞.HereBk:={u∈Ek| u ≤Rk}.
(F) For everyk∈N,there existrk∈(,Rk)anddk< such thatI(u)≤dkfor everyu∈ Ek:=
j≤kE(j)withu=rk.
(F) Isatisfies(PS)∗condition with respect to{Em|m∈N},i.e.,every sequenceum∈Em
withI(um) < bounded and(I|Em)(um)→ asm→ ∞ has a subsequence which
Then, for each k≥k,I has a critical value ξk∈[bk,dk],hence ξk < andξk→ as k→ ∞.
LetE(j) =span{ej}for eachj∈N, where{ej:j∈N}is the system of eigenfunctions given in Remark . Now we show that the functionalIhas the geometric property of Lemma under the conditions of Theorem .
Lemma Assume that(L), (L), (W),and(W)hold.Then there exist a positive integer
kand a sequence Rk→+as k→ ∞such that
inf u∈Ek,u=Rk
I(u)≥, ∀k≥k
and
bk:= inf u∈Bk
I(u)→ as k→ ∞,
where Ek:=j≥kE(j)and Bk:={u∈Ek| u ≤Rk}for all k∈N.
Proof Note thatEk⊂E+for allk≥ ¯j+ (see Remark for details). Thus for eachk≥ ¯j+ , by () we obtain
I(u)≥ u
–
R
W(t,u)dt
≥ u
–c
uθθ++–βupp, ∀u∈Ek. ()
Set
lk= sup u∈Ek,u=
uθ+, ∀k≥ ¯j+ . ()
SinceEis compactly embedded intoLθ+, we have (see [])
lk→+ ask→ ∞. ()
For eachk≥ ¯j+ , it follows from (), (), (), and the choice ofβthat
I(u)≥ u
–c
lθk+uθ+–βγppup
≥ u
–c lθk+u
θ+–
u
p, ∀u∈Ek. ()
For eachk≥ ¯j+ , choose
Rk= clθk+, ()
then, by (),
and hence there exists a positive integerk≥ ¯j+ such that
which combined with () and () implies that
bk:= inf u∈Bk
I(u)→ ask→ ∞.
The proof is completed.
and let
dk= –r
k < .
Ifu∈Ekwithu=rk, we have
I(u)≤dk.
The proof is completed.
Lemma Assume that(L), (L), (W), (W),and(W)hold.Then I satisfies(PS)∗
con-dition with respect to{Em|m∈N}.
Proof Letum∈Embe a (PS)∗sequence, that is,
I(um) is bounded and (I|Em)(um)→ asm→ ∞. ()
Then we claim that{um}is bounded. If not, passing to a subsequence if necessary, we may assume that
um → ∞ asm→ ∞. ()
From (), (), (), we have
I(um) –(I|Em)(um),um=
R
∇W(t,um),um– W(t,um)dt
≥(p– )β
{t∈R||um(t)|≥r}|um|
pdt ()
for allm∈N. From (), (), and (), it follows that
{t∈R||um(t)|≥r}|um|pdt
um → ()
asm→ ∞. Let
vm(t) =
um(t), if|um(t)|< r, , if|um(t)| ≥r
()
and
wm(t) =um(t) –vm(t) ()
for allm∈Nand allt∈R. By (), (), and (),
for all positive integerm. From Hölder’s inequality, (), (), and the equivalence of the norms on the finite-dimensional subspaceE–⊕E, we have
u–m+um =u–m+um,um
=u–m+um,vm+u–m+um,wm ≤u–m+umvm∞+u–m+umpwmp
≤cu–m+um +wmp ()
for allm∈N, where p +p = . Then, from the equivalence of the norms on the
finite-dimensional subspaceE–⊕E, (), and () it follows that
u–m+um≤cu–m+um ≤c +wmp ≤c +ump
for allm∈N, which implies that
u–
m+um um
→ ()
asm→ ∞. By () we get
∇W(t,x)≤c +|x|p–, ∀(t,x)∈R×RN,
which combined with () implies that
(I|Em)(um),u+ m
≥u+
m
–
R
∇W(t,um)u+mdt
≥u+m–c
R|um|
p–u+
mdt–c
R
u+mdt
≥u+m–cu+m∞
{t∈R||um(t)|≥r}
|um|p–dt
–c(r)p–
{t∈R||um(t)|<r}
u+mdt–cu+m
≥u+m–cu+m∞(r)–
{t∈R||um(t)|≥r} |um|pdt
–c(r)p–u+m–cu+m ≥u+m–cγ∞u+m(r)–
{t∈R||um(t)|≥r} |um|pdt
–c(r)p–γu+m–cγu+m. From this and () it follows that
u+
m
asm→ ∞. Combining () and (), one gets
asm→ ∞, which is a contradiction. Hence{um}is bounded. Noting that by Lemma ,I is a compact perturbation of the identity inEE. SinceEhas finite dimension,{um}
has a subsequence converging to a critical point ofI(see []). Hence,Isatisfies the (PS)∗
condition. The proof is completed.
Proof of Theorem It follows from Lemmas - that the functionalI satisfies the con-ditions (F)-(F) of Lemma . Therefore, by Lemma , there exists a sequence of critical
valuesξk< withξk→ ask→ ∞. Let{uk}be a sequence of critical points ofI corre-large enough, they are homoclinic solutions of problem (). The proof is completed.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This research was supported by National Natural Science Foundation of China (No. 11426190).
Received: 22 April 2015 Accepted: 18 November 2015 References
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