Abstract
Three periodic solutions with prescribed wavelength for a class of semilinear fourth-order differential inclusions are obtained by using a nonsmooth version critical point theorem. Some results of previous related literature are extended.
1. Introduction
The Fisher-Kolmogorov (FK for short) equation was proposed as a model for phase transitions and other bistable phenomena and is one of the most fundamental models in mathematical biology and ecology; for example, see Zimmerman [1], Coullet et al. [2], and Dee and SaarLoos [3].
Recently, many authors were interested in extended FK equation of the form where is a positive constant and and are continuous positive -periodic functions on . Equation (1) arises as the mesoscopic model of a phase transition in a binary system near the Lipschitz point [4, 5], and (1) is frequently used as a model for the study of pattern formation in an unstable spatially homogeneous state [2, 3]. It has been attracting more and more attention due to its significant value in theory and practical application [6–9].
Especially, in [10], by using the symmetric mountain-pass theorem, Ma and Dai considered the nonlocal semilinear fourth-order differential equation They obtained the existence of infinitely many distinct pairs of solutions of the above problem, where is a positive constant, is a positive continuous even and -periodic function, and is continuous and monotone decreasing.
It is worth mentioning that the method used in [10] is not valid for more general nonlinearity. Furthermore, to the best of our knowledge, there is no author using the nonsmooth version critical point theory to consider the extended FK equation. In view of this, this paper is concerned with the existence of three periodic solutions of the following semilinear fourth-order differential inclusion: where(H1) are real parameters, is a positive constant, and is a positive continuous even -periodic function on ; is a locally Lipschitz function defined on satisfying(F1), for ,(F2)there exist constants and , such that (F3)where and are two given positive constants.
is defined on satisfying(G1) is measurable for each , is locally Lipschitz for , and and for a.e. and ;(G2)there exists constant , such that where is defined in (F2).
By applying a nonsmooth version critical point theorem [11, 12], we prove that, when and are in given interval, (3) admits at least three solutions. Moreover, we achieve an estimate of the solutions norms independent of , and . Concretely, we get the following main result.
Theorem 1. Assume that (H1) and (F1)–(F3) hold. Then there exist a nondegenerate interval and with the following property: for any and any function satisfying (G1) and (G2), there exists , such that, for , the problem (3) admits at least three solutions with norm in less than .
Remark 2. Note that the coefficient is even and -periodic, if is a solution of (3) and is its antisymmetric extension with respect to : Combining with (F1) and (G1), we can see that -periodic extension of over is a -periodic solution of the equation
The rest of this paper is arranged as follows. Section 2 contains some preliminaries of nonsmooth analysis and abstract results which are needed later. Section 3 concerns a variational method for problem (3); in the final section, we give the proof of the main result.
2. Preliminaries
We collect some basic notions and results of nonsmooth analysis, namely, the calculus for locally Lipschitz functionals developed by Clarke [13] and Motreanu and Panagiotopoulos [14].
Let be a Banach space, be its topological dual, and a functional. We recall that is locally Lipschitz (l.L.), if, for each , there exist a neighborhood of and a real number such that If is l.L. and , the generalized directional derivative of at along the direction is The generalized gradient of at is the set Then for , is a nonempty, convex, and -compact subset [13, Proposition 1]. We call that has compact gradient if maps bounded subsets of into relatively compact subsets of . We say that is a critical point of l.L. functional if .
Lemma 3 (see [14, Proposition 1.1]). If , then is l.L. and
Lemma 4 (see [14, Proposition 1.3]). Let be an l.L. functional. Then for , is subadditive and positively homogeneous and with being a Lipschitz constant for around .
Lemma 5 (see [14, Proposition 1.6]). Let be l.L. functionals. Then
Lemma 6 (see [12, Lemma 6]). Let be an l.L. functional with compact gradient. Then is sequentially weakly continuous.
In the sequel we need the following lemmas to study the existence of solutions for the problem (3). Firstly, we present a definition.
Definition 7 (see [15, Definition 2.1]). An operator is of type , if, for any sequence in and imply that .
Lemma 8 (a particular case of [12, Theorem 14]). Let be a reflexive Banach space, an interval, a sequentially weakly lower semicontinuous functional whose derivative is of type , and an l.L. functional with compact gradient, and let . Assume that Then, there exist and with the following property: for and any l.L. functional with compact gradient, there exists such that, for every , the functional admits at least three critical points in , with norms less than .
The main hypothesis of Lemma 8 is the minimax inequality (16). An easy way to have it satisfied is illustrated by the following result obtained by Ricceri [16].
Lemma 9 (see [16, Proposition 3.1]). Let be a nonempty set and functionals, and let and , such that Then for each satisfying one has that (16) with holds.
3. Variational Approach for the Problem (3)
We introduce the Banach space endowed with the norm . Obviously, is a reflexive Banach space and compactly embedded in . So there exists constant , such that .
From the positivity of and , it is easy to see that is also a norm of . Therefore, there exist two constants and such that
Definition 10. A function is said to be a weak solution of the problem (3), it is understood an element for which there corresponds to a mapping with for a.e. , and having the property that for every , and
Definition 11. A function is called a solution of the problem (3) if and where for a.e. .
Lemma 12. If is a weak solution of (3), then is a solution of (3).
Proof. Let be a weak solution of (3). Then there exists satisfying (23) for all and for a.e. . Using integration by parts, (23) becomes
so ; especially (25) holds for with ; via integration by parts, we obtain
Due to , similar to [7, Section 2], we can get .
Now we show that the boundary conditions are satisfied. From (25) and integration by parts, we can get
Since satisfies (26), we have that holds for all , which implies that ; in view of , we get . Thus is a solution of (3).
4. Proof of the Main Result
In order to prove the main result, we shall show some related lemmas. Firstly, define functionals
Next, we will give some properties of those functionals.
Lemma 13. Assume (H1) holds; then and is of type .
Proof. Clearly, by (H1) and the definition of , we know and Let such that and . According to Definition 7, we shall show that . In fact On the other hand, owing toEquation (32) subtracting (31), we get so It follows from the equivalence of and that as .
Lemma 14. Assume (H1), (F1), (F2), (G1), and (G2) hold. Then the functional is l.L. Moreover, for each critical point of , is a weak solution of (3).
Proof. Let , where . Due to , by Lemma 3, is l.L. on . From (F2) and (G2), we know that is l.L. on . Moreover is compactly embedded into , so is also l.L. on ([17, Theorem 2.2] and [18]). Furthermore, according to Lemma 5, we get
The interpretation of (35) is as follows: for every , there corresponds a mapping for a.e. having the property that, for every , the function and . Therefore is l.L. on .
Now we show that each critical point of is a weak solution of (3). Let be a critical point of . So
By Lemma 3 and (36), we get
and hence a.e. on ; it follows from (30) and (37), that for every ; we have
Thus by Definition 10, is a weak solution of (3).
Lemma 15. Suppose (F1) and (F2) are fulfilled. Then is compact.
Proof. According to Lemma 14, is l.L. on and . Let such that for every ; choose for a.e. , , and ; then by (F2), we get for every , ,
where and are positive constants, so and the sequence is bounded, and hence, up to a subsequence if necessary, .
We claim that .
Suppose the contrary; we assume there exists such that (choose a subsequence if necessary). Hence for every , we can find with such that
Passing to a subsequence if necessary, we can assume that in , so in and . Combining with (F2), we have
which contradicts (40).
Proof of Theorem 1. The equivalence of defined in (21) with the usual norm in implies that the functional defined in (28) is sequentially weak lower semicontinuous. By Lemma 13; is of type . By Lemmas 14 and 15, is an l.L. functional with compact gradient.
First, we verify the condition (15) in Lemma 8. By (F1), (F2), and Lebourg’s mean value theorem, we have
hence
where and are positive constants, so it follows from that
Next, we verify the condition (16) by using Lemma 9. Set , . Clearly, and . Moreover, if we choose , where is small enough, then
so (18) holds. For with , considering the compact embedding of in , we have
So, we obtain
Choose . In view of (F3) and (47), we have
and therefore, (19) holds. Then we get (16) for some and .
For function which satisfies (G1) and (G2), it follows from Lemma 14 that the functional is l.L. Similar to the argument of Lemma 15, we can obtain that is compact. Then according to Lemma 8, there exist and with the property that, for and the above , there is such that, for , the functional
admits at least three critical points with , . So by Lemmas 14 and 12, , , are three solutions of the problem (3).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Program for New Century Excellent Talents in University (NECT-12-0246) and FRFCU (lzujbky-2013-k02).