# Infinitely Many Homoclinic Solutions for Nonperiodic Fourth Order Differential Equations with General Potentials

Authors: Liu Yang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We investigate a class of nonperiodic fourth order differential equations with general potentials. By using variational methods and genus properties in critical point theory, we obtain that such equations possess infinitely homoclinic solutions.


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## Body

## 1. Introduction

In this paper, we consider the following a class of fourth order differential equations:(1)u(4)+wu′′+a(x)u=f(x,u),x∈R,
where w is a constant, f∈C(R×R,R), and a∈C(R,R).Recently, a lot of attention has been focused on the study of homoclinic and heteroclinic solutions for this problem; see [1–8]. This may be due to its concrete applications, such as physics and mechanics; see [9, 10]. More precisely, Tersian and Chaparova [5] studied periodic case. They obtained nontrivial homoclinic solutions by using mountain pass theorem. For nonperiodic case, Li [7] studied the existence of nontrivial homoclinic solutions for this class of equations. Sun and Wu [8] studied multiple homoclinic solutions for the following nonperiodic fourth order equations with a perturbation:
(2)u(4)+wu′′+a(x)u=f(x,u)+λh(x)|u|p-2u,0000000000000000000000x∈R,1≤p<2.
By using the mountain pass theorem and Ekeland variational principle, two homoclinic solutions for these equations are obtained under the assumption that A0 and f are superlinear or asymptotically linear as |u|→+∞, where (A0) is the following condition: (A0)
there exists a positive constanta1 such that 0<a1≤a(x)→+∞ as |x|→+∞ and w≤2a1.The assumption(A0) is too strict to be satisfied by many general functions a(x); for example, a(x)=1. In addition, although there is perturbation, the right of (2) is superlinear or asymptotically linear as |u|→+∞. In the present paper we study the infinitely many homoclinic solutions for (1) under more general assumption than A0 and sublinear condition on f.Before stating our results we introduce some notations. Throughout this paper, we denote by∥∥r the Lr-norm, 2≤r≤∞. L∞(R) is the Banach space of essentially bounded functions equipped with the norm
(3)∥u∥∞=esssup⁡{|u(x)|:x∈R}.
If we take a subsequence of a sequence {un} we will denote it again by {un}.Now we state our main result.Theorem 1.
Assume that the following conditions hold:(A) there exists a positive constant a1 such that a(x)≥a1>0 and 2a1≥w;(F1) there exists a constant 1<γ<2 and positive function b(x)∈L2/(2-γ)(R,R+) such that
(4)|f(x,u)|≤γb(x)|u|γ-1,∀(x,u)∈R×R;(F2) there exist x0∈R and ν∈(1,2) such that
(5)liminf⁡(x,u)→(x0,0)F(x,u)|u|ν>0,where F is the primitive F(x,u)=∫0u‍f(x,t)dt.
Then, problem (1) possesses at least one nontrivial homoclinic solution.
In addition, iff is odd symmetry in u, that is,(F3)f(x,u)=-f(x,-u), ∀(x,u)∈R×R,then one gets the existence of infinitely many nontrivial homoclinic solutions.Theorem 2.
Under the assumptions of(A),(F1)–(F3), problem (1) possesses infinitely many nontrivial homoclinic solutions.Example 3.
Iff(x,u)=(4cos⁡x/3e|x|)|u|-2/3u, clearly,
(6)|f(x,u)|≤43e|x||u|1/3u,∀(x,u)∈R×R,F(x,u)=cos⁡xe|x||u|4/3≥cos⁡1e|u|4/3,∀(x,u)∈[0,1]×R.
Thus (F1), (F2), and (F3) are satisfied with x0=0,ν=4/3,γ=4/3.The paper is organized as follows. In Section2, we present some preliminaries. In Section 3, we give the proof of our main results.


## 2. Preliminaries

In order to prove our main results, we first give some properties of spaceX on which the variational setting for problem (1) is defined.Lemma 4 (see [5]).
Assume that(A) hold. Then there exists a constant c0>0 such that
(7)∫R‍[u′′(x)2-wu′(x)2+a(x)u(x)2]dx≥c0∥u∥H22,00000000000000000000000000000∀u∈H2(R),
where ∥u∥H22=(∫R‍[u′′(x)2+u′(x)2+u(x)2]dx)1/2 is the norm of Sobolev space H2(R).
Letting(8)X={u∈H2(R)∣∫R[u′′(x)2-wu′(x)200+a(x)u(x)2]dx<+∞∫R[u′′(x)2-wu′(x)2},
then X is a Hilbert space with the inner product
(9)(u,v)=∫R‍[u′′(x)v′′(x)-wu′(x)v′(x)+a(x)u(x)v(x)u′′(x)]dx
and the corresponding norm ∥u∥2=(u,u). Note that
(10)X⊂H2(R)⊂Lr(R),
for all r∈[2,+∞], with the embedding being continuous. Hence, for any r∈[2,+∞], there is Cr>0 such that
(11)∥u∥r≤Cr∥u∥,∀u∈X.Now we begin describing the variational formulation of problem (1). Consider the functional J:X→R defined by
(12)J(u)=12∥u∥2-∫R‍F(x,u)dx.Lemma 5.
Under the conditions of(A),(F1)–(F2), J∈C1(X,R) and its derivative is given by the following;
(13)J′(u)v=∫R‍[u′′(x)v′′(x)-wu′(x)v′(x)+a(x)u(x)v(x)]dx-∫R‍f(x,u(x))v(x)dx,
for all u,v∈X. In addition, any critical point of J on X is a classical solution of problem (1).Proof.
We firstly show thatJ:X→R. From (F1), one has
(14)|F(x,u)|≤b(x)|u|γ,∀(x,u)∈R×R.
By the Hölder inequality and (14), we have
(15)∫R‍|F(x,u(x))|dx≤∫R‍b(x)|u(x)|γdx≤∥b∥2/(2-γ)(∫R|u(x)|2dx)γ/2≤C2γ∥b∥2/(2-γ)∥u∥γ,
where C2 is constant in (11). Hence, J defined by (12) is well defined on X.
Next we prove thatJ∈C1(X,R). To this end, we rewrite J as follows:
(16)A(u)=12∥u∥2,B(u)=∫R‍F(x,u)dx.
It is easy to check that A∈C1(X,R), and we have A′(u)v=∫R‍[u′′(x)v′′(x)-wu′(x)v′(x)+a(x)u(x)v(x)]dx, for all u,v∈X. On the other hand, we will show that B∈C1(X,R) and
(17)B′(u)v=∫R‍f(x,u(x))v(x)dx,
for any given u,v∈X. For any given u∈X, let us define
(18)K(u)v=∫R‍f(x,u(x))v(x)dx,∀v∈X.
It is obvious that K(u) is linear. Now we show that K(u) is bounded. In fact, for any u∈X, by the Hölder inequality and (F1), we can obtain that
(19)|K(u)v|≤∫R|f(x,u(x))v(x)|dx≤γ∫Rb(x)|u(x)|γ-1|v(x)|dx≤γ∥b∥2/(2-γ)(∫R|u(x)|2dx)(γ-1)/2(∫R|v(x)|2dx)1/2≤γC2γ∥b∥2/(2-γ)∥u∥γ-1∥v∥.
Moreover, foru,v∈X, using the Mean Value Theorem, we get
(20)∫R‍F(x,u(x)+v(x))dx-∫R‍F(x,u)dx=∫R‍f(x,u(x)+θ(x)v(x))v(x)dx,
for some 0<θ(x)<1. On the other hand, b(x)∈L2/(2-γ)(R,R+), for any ε>0, there exists T>0 such that
(21)(∫|x|>Tb2/(2-γ)(x)dx)(2-γ)/2<ε.
Therefore, on account of the Sobolev compact theorem (X|[-T,T] is compactly embedded in L∞([-T,T],R)) and Hölder inequality, we have
(22)1∥v∥[∫R‍|f(x,u(x)+θ(x)v(x))v(x)-f(x,u(x))v(x)|dx]≤1∥v∥[∫|x|≤T|f(x,u(x)+θ(x)v(x))v(x)-f(x,u(x))v(x)|dx∫|x|≤T|f(x,u(x)+θ(x)v(x))v(x)]+1∥v∥[∫|x|>T|f(x,u(x)+θ(x)v(x))v(x)-f(x,u(x))v(x)|dx∫|x|>T|f(x,u(x)+θ(x)v(x))v(x)]≤1∥v∥[∫|x|≤T|f(x,u(x)+θ(x)v(x))-f(x,u(x))|2dx∫|x|≤T|f(x,u(x)+θ(x)v(x))]1/2×[∫|x|≤T|v(x)|2dx]1/2+2∥v∥[∫|x|>Tγb(x)[|u(x)|γ-1+|v(x)|γ-1]|v(x)|dx]≤1∥v∥[∫|x|≤T|f(x,u(x)+θ(x)v(x))-f(x,u(x))|2dx∫|x|≤T|f(x,u(x)+θ(x)v(x))]1/2C2∥v∥+2γ(∫|x|>T‍|b(x)|2/(2-γ)dx)(2-γ)/2×(∫|x|>T|u(x)|2dx)(γ-1)/2(∫|x|>T|v(x)|2dx)1/2+γ(∫|x|>T|b(x)|2/(2-γ)dx)(2-γ)/2×(∫|x|>T|v(x)|2dx)γ/2≤C2[∫|x|≤T|f(x,u(x)+θ(x)v(x))-f(x,u(x))|2dx]1/2+εγC2γ(∥u∥γ-1+∥v∥γ-1)⟶0,asv⟶0,
which, together with (19), implies that (17) holds. It remains to show that B′ is continuous. Suppose that u→u0 in X, then we have
(23)sup⁡∥v∥=1|B′(u)v-B′(u0)v|=sup⁡∥v∥=1[∫R|f(x,u(x))v(x)-f(x,u0(x))v(x)|dx]≤sup⁡∥v∥=1[∫|x|≤T|f(x,u(x))v(x)-f(x,u0(x))v(x)|dx]+sup⁡∥v∥=1[∫|x|>T|f(x,u(x))v(x)-f(x,u0(x))v(x)|dx]≤sup⁡∥v∥=1[∫|x|≤T‍|f(x,u(x))-f(x,u0(x))|2dx]1/2×[∫|x|≤T‍|v(x)|2dx]1/2+sup⁡∥v∥=1[∫|x|>Tγb(x)[|u(x)|γ-1+|u0(x)|γ-1]|v(x)|dx]≤sup⁡∥v∥=1[∫|x|≤T|f(x,u(x))-f(x,u0(x))|2dx]1/2C2∥v∥+γsup⁡∥v∥=1(∫|x|>T‍|b(x)|2/(2-γ)dx)(2-γ)/2×(∫|x|>T|u(x)|2dx)(γ-1)/2(∫|x|>T‍|v(x)|2dx)1/2+γsup⁡∥v∥=1(∫|x|>T|b(x)|2/(2-γ)dx)(2-γ)/2×(∫|x|>T|u0(x)|2dx)(γ-1)/2×(∫|x|>T|v(x)|2dx)1/2≤C2[∫|x|≤T|f(x,u(x))-f(x,u0(x))|2dx]1/2+εγC2γ(∥u∥γ-1+∥u0∥γ-1)⟶0,asu⟶u0,
uniformly with respect to v, which implies that B′ is continuous. Therefore, we obtain J∈C1(X,R) and its derivative is given by
(24)J′(u)v=∫R[u′′(x)v′′(x)0-wu′(x)v′(x)+a(x)u(x)v(x)]dx-∫R‍f(x,u(x))v(x)dx,
for all u,v∈X. In addition, from [8], we can know that any critical point of J on X is a classical solution of problem (1).Next, we give some useful Lemmas which can be seen in [11].Definition 6.
I
∈
C
1
(
E
,
R
) is said to satisfy the (PS) condition if any sequence {un}⊂E,n∈N, for which {I(un)} is bounded and I′(un))→0 as n→+∞ and possesses a convergent subsequence in E.Lemma 7.
LetE be a real Banach space and let I∈C1(E,R) satisfy the (PS) condition. If I is bounded from below, then c=inf⁡EI(u) is a critical value of I.To obtain the existence of infinitely many homoclinic solutions for problem (1) under the assumptions of Theorem 2, we will employ the “genus” properties in critical point theory; see [11].LetE be a Banach space, I∈C1(E,R), and c∈R. We set
(25)Σ={A⊂E-{0}:0Ais closed inEand symmetric with respect to0},Kc={u∈E:I(u)=c,I′(u)=0},Ic={u∈E:I(u)≤c}.Definition 8.
ForA∈Σ, we say the genus of A is J (denoted by γ(A)=j) if there is an odd map ψ∈C((A,R)j∖{0}) and j is the smallest integer with this property.Lemma 9.
LetI be an even C1 functional on E and satisfy the (PS) condition. For any j∈N, set
(26)Σj={A∈Σ:γ(A)≥j},cj=inf⁡A∈Σjsup⁡u∈AI(u).(i)
IfΣj≠∅ and cj∈R, then cj is a critical value of I;(ii)
if there existsr∈N such that
(27)cj=cj+1=⋯=cj+r=c∈R,and c≠I(0), then γ(Kc)≥r+1.Remark 10.
From Remark 7.3 in [11], we know that if Kc∈Σ and γ(Kc)>1, then I has infinitely many distinct critical points in E.


## 3. Proof of the Main Results

To prove our main results, we first give the following Lemma.Lemma 11.
If(A) and (F2) hold, then J defined by (12) satisfies (PS) condition.Proof.
By (12) and (14) and Hölder inequality, one has
(28)J(u)=12∥u∥2-∫R‍F(x,u)dx≥12∥u∥2-∫R‍b(x)|u(x)|γdx≥12∥u∥2-∥b∥2/(2-γ)(∫R‍|u(x)|2dx)γ/2≥12∥u∥2-C2γ∥b∥2/(2-γ)∥u∥γ.
Since 1<γ<2, (28) implies that J(u)→+∞ as ∥u∥→+∞. Consequently, J is bounded from below.
Now, we show thatJ satisfies the (PS) condition. Assume that {uk}k∈N⊂X is a sequence such that {J(uk)}k∈N is bounded and J′(uk)→0 as k→+∞. Then by (28), there exists a constant C>0 such that
(29)∥uk∥2≤C2∥uk∥≤C,k∈N.
So passing to a subsequence if necessary, it can be assumed that uk⇀u0 in X. Since b(x)∈L2/(2-γ)(R,R+), for any ε>0, there exists T>0 such that
(30)(∫|x|>T‍b2/(2-γ)(x)dx)(2-γ)/2<ε.
Since the embedding of X↪Lloc⁡2(R) is compact, uk⇀u0 in X implies
(31)lim⁡k→∞∫|x|≤T‍|uk(x)-u0(x)|2dx=0.
Then we have
(32)∫R|f(x,uk(x))-f(x,u0(x))||uk(x)-u0(x)|dx≤[∫|x|≤T‍|f(x,uk(x))-f(x,u0(x))||uk(x)-u0(x)|dx]+[∫|x|>T|f(x,uk(x))-f(x,u0(x))|×|uk(x)-u0(x)|dx∫|x|>T|f(x,uk(x))-f(x,u0(x))|]≤[∫|x|≤T|f(x,uk(x))-f(x,u0(x))|2dx]1/2×[∫|x|≤T‍|uk(x)-u0(x)|2dx]1/2+∫|x|>Tγb(x)[|uk(x)|γ-1+|u0(x)|γ-1]×[|uk(x)|+|u0(x)|]dx≤[∫|x|≤T|f(x,uk(x))-f(x,u0(x))|2dx]1/2×[∫|x|≤T|uk(x)-u0(x)|2dx]1/2+2γ∫|x|>Tb(x)[|uk(x)|γ+|u0(x)|γ]dx≤[∫|x|≤T|f(x,uk(x))-f(x,u0(x))|2dx]1/2×[∫|x|≤T‍|uk(x)-u0(x)|2dx]1/2+2γ[∫|x|>T‍|b(x)|2/(2-γ)](2-γ)/2[∥uk∥2γ+∥u0∥2γ]≤[∫|x|≤T|f(x,uk(x))-f(x,u0(x))|2dx]1/2×[∫|x|≤T‍|uk(x)-u0(x)|2dx]1/2+2γ[∫|x|>T‍|b(x)|2/(2-γ)](2-γ)/2[Cγ+∥u0∥2γ].
Hence, by (30), (31), and the fact that ε is arbitrary, one can get
(33)∫R‍|f(x,uk(x))-f(x,u0(x))||uk(x)-u0(x)|dx⟶0
as k→∞. It follows from (13) that
(34)〈J′(uk)-J′(u0),uk-u0〉=∥uk-u0∥2-∫R|f(x,uk(x))-f(x,u0(x))||uk(x)-u0(x)|dx.
In view of the definition of weak convergence, we have
(35)〈J′(uk)-J′(u0),uk-u0〉⟶0.
Therefore, we can obtain that uk→u0 in X. Hence, J satisfies (PS) condition.Now we are in the position to complete the proof of Theorems1 and 2.Proof of Theorem1.
It is obvious thatJ(0)=0, and by Lemmas 5 and 11 we know that J is a C1 functional on X satisfying the (PS) condition. In view of (28), we have J is bounded below on X. Hence, by Lemma 7, c=inf⁡EJ(u) is a critical value of J; that is, there exists a critical point u*∈X such that J(u*)=c.
In addition, by(F2), there exists an open set D⊂R with x0∈D, σ>0, η>0 such that
(36)F(x,u)≥η|u|ν,∀(x,u)∈D×R,|u|≤σ.
Let u0∈W02,2(D)∩X∖{0} and ∥u0∥∞≤σ; then we have
(37)J(su0)=12s2∥u0∥2-∫R‍F(x,su0)dx=12s2∥u0∥2-∫D‍F(x,su0)dx≤12s2∥u0∥2-ηsν∫D‍|u0|νdx,
where 0<s<1. Since 1<ν<2, one has J(su0)<0, for s>0 small enough. Hence, u*≠0, J(u*)<0; therefore u* is a nontrivial homoclinic solution for problem (1).Proof of Theorem2.
Now, by(F3), we have J is even and J(0)=0. In order to apply Lemma 9, we prove that there exists ε>0 such that
(38)γ(J-ε)≥j,
for any j∈N. For any j∈N, we take j disjoint open sets Di such that ∪i=1jDi⊂D. For i=1,2,…,j, let ui∈W02,2(Di)∩X∖{0} with ∥ui∥=1, and
(39)Ej=span⁡{u1,u2,…,uj},Sj={u∈Ej:∥u∥=1}.
Then, for any u∈Ej, there exist λi∈R, i=1,2,…,j such that
(40)u(x)=∑i=1j‍λiui(x).
Then,
(41)∥u∥ν=(∫R|u(x)|νdx)1/ν=(∑i=1jλiν∫Di|u(x)|νdx)1/ν,(42)∥u∥2=∫R[u′′(x)2-wu′(x)2+a(x)u(x)2]dx=∑i=1j‍λi2∫Di[ui′′(x)2-wui′(x)2+a(x)ui(x)2]dx=∑i=1j‍λi2∫R[ui′′(x)2-wui′(x)2+a(x)ui(x)2]dx=∑i=1j‍λi2∥ui∥2=∑i=1j‍λi2.
Since all norms of a finite dimensional norm space are equivalent, so there exists a constant d>0 such that
(43)d∥u∥≤∥u∥ν,∀u∈Ej.
For all u∈Sj and sufficient small s>0, we have
(44)J(su)=12s2∥u∥2-∫R‍F(x,su)dx=12s2∥u∥2-∑i=1j‍∫Di‍F(x,sλiui(x))dx≤12s2∥u∥2-ηsν∑i=1j‍|λi|ν∫Di‍|ui|νdx≤12s2∥u∥2-ηsν∥u∥νν≤12s2∥u∥2-η(ds)ν∥u∥ν≤12s2-η(ds)ν.
In this case (36) is applicable, since u is continuous on D- and so |sλiui(x)|≤σ, ∀x∈D, i=1,2,…,j can be true for sufficiently small s. Therefore, it follows from (44) that there exist ε>0 and δ>0 such that
(45)J(δu)<-εforu∈Sj.
Let
(46)Sjδ={δu:u∈Sj},Ω={(λ1,λ2,…,λj)∈Rj:∑i=1j‍λi2<δ2}.
Then, it follows from (45) that
(47)J(u)<-ε,∀u∈Sjδ,
which together with the fact that J is even C1 functional on X, yields that
(48)Sjδ⊂J-ε∈Σ,
where J-ε and Σ have been previously introduced in Section 2. On the other hand, it follows from (40) and (42) that there exists an odd homeomorphism ψ∈C(Sjδ,∂Ω). By some properties of the genus (see 3∘ of Propositions 7.5 and 7.7 in [11]), we infer
(49)γ(J-ε)≥γ(Sjδ)=j,
so the proof of (38) follows. Set
(50)cj=inf⁡A∈Σjsup⁡u∈AJ(u),
where Σj is defined in Lemma 9. It follows from (50) and the fact that J is bounded from below in X that we have -∞<cj≤-ε<0, which implies that, for any j∈N, cj is a real negative number. By Lemma 9 and Remark 10, J has infinitely many nontrivial critical points, and consequently, problem (1) possesses infinitely many nontrivial homoclinic solutions.



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