Abstract
We consider the existence of infinitely many classical solutions to a class of impulsive differential equations with Dirichlet boundary value condition. Our main tools are based on variant fountain theorems and variational method. We study the case in which the nonlinearity is sublinear. Some recent results are extended and improved.
1. Introduction
Consider the following Dirichlet boundary value problem of impulsive differential equations: where , , , is continuous, and , are continuous.
Since impulsive differential equations can describe many evolution processes in which their states are changed abruptly at certain moments of time, they play an important role in applications, such as in control theory, optimization theory, biology, and some physics or mechanics problem; see [1–5]. For general theory of impulsive differential equations, we refer the readers to the monographs as [6, 7]. The existence and multiplicity of solutions to impulsive differential equations with boundary value condition have been obtained by using fixed point theorems and upper and lower solutions method; see [8–12] and references therein. Recently, some authors creatively applied variational method to deal with impulsive problems and obtained some new results; see [13–18]. For general theory of variational method, we refer the readers to the monographs as [19, 20]. More precisely, Nieto and O'Regan [13] studied Dirichlet problem as follows: For the sublinear case, they obtained the following result.
Lemma 1 (See [13]). Assume that the following conditions are satisfied. (1)There exist and such that (2)There exist and () such that Then Problem (2) has at least one solution.
For superlinear case, Zhang and Yuan [15] obtained the existence of one and infinitely many solutions for Problem (2) under the well-known Ambrosetti-Rabinowitz condition; that is, there exists such that where is a primitive function of . Soon after, Zhou and Li [16] obtained the existence of infinitely many solutions for Problem (1) under the weaker condition; there exist and such that Recently, Sun and Chen studied the existence of infinitely many solutions for Problem (1) with superlinear nonlinearity which is not satisfied (5) or (6). In addition, they also studied the case where the nonlinearity is asymptotically linear.
Motivated by the above facts, in this paper, our aim is to study the existence of infinitely many solutions for Problem (1) with nonlinearity which is sublinear. To the best of our knowledge, there are few papers concerned with this. For sublinear case, Nieto and O'Regan [13] only obtain the existence of at least one solution.
We make the following assumptions:() are odd and satisfy for all .()For any , there exist constants and such that for all .()There exist constants , and such that for every and with , where . Moreover, for all and .()There exist constants and such that . In what follows, , denote positive constants.()there exist constants , , such that for every and .() is even in , that is, .
Theorem 2. Assume that are satisfied. Then Problem (1) has infinitely many classical solutions.
Remark 3. For the definition of classical solution, we refer readers to paper [17, 18]. By and , () are not sublinear as those in [13–18]. We note that there are functions () and which satisfy the conditions of Theorem 2 but do not satisfy the conditions in references we mentioned above. For example, let If we choose , , , , , and , then it is easy to check that the conditions in Theorem 2 are satisfied.
The organization of this paper is as follows. In Section 2, we shall give some lemmas and some preliminary results. In Section 3, the proofs of the main results are given.
2. Preliminaries
In order to prove our main results, we recall the variant fountain theorem. Let be a Banach space with the norm and with for any . Set , . Consider the following -functional defined by
Lemma 4 (see [21]). Suppose that the functional defined above satisfies the following. (C1) maps bounded sets to bounded sets uniformly for . Furthermore, for all .(C2) as on any finite dimensional subspace of .(C3)There exist such that for all and as uniformly for . Then there exist such that , as . In particular, if has a convergent subsequence for every , then has infinitely many nontrivial critical points satisfying as .
In the Sobolev space , consider the inner product inducing the norm for any . Since is compactly embedded in with norm for , as in [18, 22], we know that the eigenvalues of operator with the Dirichlet boundary conditions are numbered by (counted in their multiplicities) and a corresponding system of eigenfunctions , which forms the completely orthogonal basis in . Assume , and let , , and . Then . We introduce on the following product and norm , where , . Then and are equivalent. By the Soblev imbedding theorem, is compactly embedded in , and there exists such that where
Define a functional on by where with , , . By the conditions of Theorem 2, we know that is continuously differentiable and for any .
Like for Lemma 2.4 in [17], one can prove that the critical points of the functional are the classical solutions for Problem (1).
3. Proofs of the Main Results
Now we define a class of functionals on by where , . Denote by , . Clearly, we see that for all and the critical points of correspond to the solutions to Problem (1).
Lemma 5. Let be satisfied. Then . Furthermore, as on any finite dimensional subspace of .
Proof . Evidently, follows by the definition of the functional and . Now we claim that for any finite dimensional subspace of , there exists such that
where meas denotes the Lebesgue measure in .
Otherwise, for any , there exists such that
Let for all . Then and
Since , it follows from the compactness of the unit sphere of that there exists a subsequence, say , such that in . Hence, we have . By the equivalence of the norms on the finite dimensional space , we have in , that is,
Thus there exist such that
In fact, if not, we have
This implies that
which gives a contradiction.
Now let
and . We have
for all positive integer . Let be large enough such that and . Then we have
This implies that
for all large , which is a contradiction with (24). For the given in (21), let
Then
Observing that for any with , the following inequality holds
Combining (34) and , for any with , we have
This implies as on any finite dimensional subspace of .
Lemma 6. Assume that , are satisfied. Then there exist a positive integer and a sequence as such that and as uniformly for , where for all .
Proof . Note first that for all by definition of in Section 2. Thus for any , by , , we have for all . Set . Then Indeed, it is clear that , so as . For every , there exists such that and . By the definition of , in . Then this implies that in . Thus we have proved that . Therefore, for any , the following inequality holds: for all . Let . Then by (38), we have as . Hence, by (39), straightforward computation shows that . Furthermore, for any and with , we have Clearly, we see that by the definition of and . Therefore, Combining (38) and (41), we have as uniformly for .
Lemma 7. Let , be satisfied. Then for the sequence obtained in Lemma 6, there exist for all such that where for all .
Proof . Let with . By (16), for any with , one has . By , we obtain for any with , where the last inequality follows by the equivalence of the norms on the finite dimensional space . Since , (), for , are small enough, we can get , .
Proof of Theorem 2. By , , maps bounded sets to bounded sets uniformly for . Evidently, , imply that for all . Thus by Lemma 4, there exist , such that , as . For the sake of notational simplicity, in what follows we always set for all . By , , one has where and is a constant. Hence, there exists a constant such that , . On the other hand, we can easily obtain that , . Thus, we have . In view of the equivalence of any two norms on finite dimensional space and (16), we obtain where ( when . Therefore, we have In view of the equivalence of norms on , we obtain , . Note that Thus Since , we have ; that is, is bounded in . By a standard argument, this yields a critical point of such that . Since as , we can obtain infinitely many critical points.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (no. 10871206).