Multiplicity of Homoclinic Solutions for a Class of Nonperiodic Fourth-Order Differential Equations with General Perturbation
In this paper, we investigate a class of nonperiodic fourth-order differential equations with general perturbation. By using the mountain pass theorem and the Ekeland variational principle, we obtain that such equations possess two homoclinic solutions. Recent results in the literature are generalized and significantly improved.
In this paper, we consider the following class of fourth-order differential equations: where is a constant, , , and .
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  studied periodic case. They obtained nontrivial homoclinic solution by using mountain pass theorem. For nonperiodic case, Li  studied the existence of nontrivial homoclinic solution for this class of equations. Sun and Wu  studied multiple homoclinic solutions for the following nonperiodic fourth-order equations with a perturbation: Before stating the results of  and our results, we introduce some notations. Throughout this paper, we denote by the -norm, . is the Banach space of essentially bounded functions equipped with the norm If we take a subsequence of a sequence , we will denote it again by .
Theorem 1 (see ). For any real number , if the following conditions are satisfied,
there exists a positive constant such that as and .
There exists with such that
There exist two constants satisfying and such that where , then one has the following results:
if and with then there exists such that, for every , problem (2) has at least two homoclinic solutions,
if , then there exists such that, for every , problem (2) has at least two homoclinic solutions.
The sublinear perturbation , is too strict; for example, does not satisfy this perturbation. However, one can see that our results in this paper can also work in this case. In addition, in Theorem 1 was defined in . Motivated by the above results, in this paper, we consider more general perturbation and the case of defined in local bounded open set .
Now, we state our main result.
Theorem 2. For any real number , if , and the following conditions are satisfied,
there exist a constant and a positive function such that there exist a constant , a bounded open set , and a positive function such that where , then one has the following results:
if and with then there exists such that, for every , problem (1) has at least two homoclinic solutions,
if , then there exists such that, for every , problem (1) has at least two homoclinic solutions.
In order to prove our main results, we first give some properties of space on which the variational setting for problem (1) is defined.
Lemma 3 (see ). Assume that the holds. Then, there exists a constant such that where is the norm of Sobolev space .
Letting then is a Hilbert space with the inner product and the corresponding norm . Note that for all with the embedding being continuous.
Lemma 4 (see ). Assume that the condition holds. Then, is compactly embedded in for all .
Now, we begin describing the variational formulation of problem (1). Consider the functional defined by Since , are continuous, by Lemma 4, and its derivative is given by for all . In addition, any critical point of on is a classical solution of problem (1).
Next, we give the variant version of the mountain pass theorem which is important for the proof of our main results.
Theorem 5 (see ). Let be a real Banach space with its dual space , and suppose that satisfies for some , and with . Let be characterized by where is the set of continuous paths joining and ; then, there exists a sequence such that
3. Proof of the Main Results
To prove our main results, we first give the following lemma.
Lemma 6. For any real number , assume that conditions , , and hold. Then, there exists such that for every there exist two positive constants such that .
Proof. For any , it follows from conditions that there exist and such that By (9) and (21), the Sobolev inequality, and the Hölder inequality, one has, for all , Take and define It is easy to prove that there exists such that Then, it follows from (22) that there exists such that for every there exist two positive constants , such that .
Consider the minimum problem Then, we have the following results.
Lemma 7. There exist a constant and with such that and that is, the minimum (25) is achieved.
Proof. For any with , by Lemma 3 and the Sobolev embedded theorem, we have Therefore, there exists a constant such that . Let be a minimizing sequence of (25). Clearly, and is bounded. Then, there exist a subsequence and such that weakly in and strongly in . So it is easy to verify that as and . Therefore, This implies that
Lemma 8. For any real number , assume that conditions , , and hold. Let be as in Lemma 6. Then, one has the following results.
If and , then there exists with such that for all .
If , then there exists with such that for all .
Proof. In case . Since , we can choose a nonnegative function with such that
Therefore, by condition and the Fatou lemma, we have
So, as ; then, there exists with such that for all .
In case . with in , and we can choose a function such that Therefore, by condition and the Fatou lemma, we have
So, as ; then, there exists with such that for all .
Then, we have the following results.
Proof. For large enough, by the Hölder inequality and Lemma 3, one has which implies that is bounded in , since .
Proof. Since with in , we can choose a function such that Hence, we have for small enough. This implies . By the Ekeland variational principle, there exists a minimizing sequence such that and as . Hence, Lemma 4 implies that there exists such that and .
Proof of Theorem 2. From Lemmas 4 and 8, there is such that, up to a subsequence, weakly in and strongly in for . By using a standard procedure, we can prove that strongly in . Moreover, and is another homoclinic solution of problem (1). Therefore, combining with Lemma 10, we prove that problem (1) has at least two homoclinic solutions satisfying and .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The paper was supported by NFSC (11271372).