Abstract
In this paper we are interested in multiplicity results for a nonlinear Dirichlet boundary value problem subject to perturbations of impulsive terms. The study of the problem is based on the variational methods and critical point theory. Infinitely many solutions follow from a recent variational result.
MSC: 34B37, 34B15.
Keywords:
impulsive differential equations; critical points; infinitely many solutions1 Introduction
The theory of impulsive differential equations provides a general framework for the mathematical modeling of many real world phenomena; see, for instance, [13] and [4]. Indeed, many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. Impulsive differential equations are basic tools for studying these phenomena [5,6].
There are some common techniques to approach these problems: the fixed point theorems [7,8], the method of upper and lower solutions [9], or the topological degree theory [1012]. On the other hand, in the last few years, some authors have studied the existence of solutions by variational methods; see [1319].
Here, we use critical point theory to investigate the existence of infinitely many solutions for the following nonlinear impulsive differential problem:
where , , , with and , , , and are continuous for every .
We establish some multiplicity results for problem () under an appropriate oscillation behavior of the primitive of the nonlinearity g and a suitable growth of the primitive of at infinity, for all λ belonging to a precise interval and provided μ is small enough (Theorem 3.3, Theorem 3.4). It is worth noticing that, when the impulsive effects , , are sublinear at infinity, our results hold for all (see Remark 3.1). Here, as an example of our results, we present the following special case of Theorem 3.3.
Theorem 1.1Letbe a continuous function and putfor every. Assume that
Then there is, where, such that for each, the problem
admits infinitely many pairwise distinct classical solutions.
We explicitly observe that in Theorem 1.1 impulsive effects , , (that is, for all ) are linear, contrary to the usual assumption of sublinearity of impulses; see [14,16,2022] and [23]. The rest of this paper is organized as follows. In Section 2, we introduce some notations and preliminary results. Moreover, the abstract critical point theorem (Theorem 2.1) is recalled. In Section 3, we obtain some existence results. In Section 4, we give some examples to illustrate our results.
2 Preliminaries
By a classical solution of () we mean a function
that satisfies the equation in () a.e. on , the limits , , , exist, that satisfies the impulsive conditions and the boundary conditions . Clearly, if a, b and g are continuous, then a classical solution , , satisfies the equation in () for all .
We consider the following slightly different form of problem ():
It is easy to see that, by choosing
the solutions of () are solutions of ().
Let us introduce some notations. In the Sobolev space , consider the inner product
which induces the norm
The following lemmas are useful for proving our main result. Their proofs can be found in [24].
Lemma 1 ([[24], Proposition 2.1])
Here, and in the sequel, is an L^{1}Carathéodory function, namely:
(b) is continuous for almost every ;
(c) for every , there exists a function such that
Definition 1 A function is said to be a weak solution of () if u satisfies
Lemma 2 ([[24], Lemma 2.1 ])
is a weak solution of () if and only ifuis a classical solution of ().
Now, we define the functionals in the following way:
for each , where for each . Using the property of f and the continuity of , , we have that and for any , one has
and
So, arguing in a standard way, it is possible to prove that the critical points of the functional are the weak solutions of problem () and so they are classical solutions.
In the next section we shall prove our results applying the following infinitely many critical points theorem obtained in [25]. First, we recall the following definition.
Definition 2 Let X be a real Banach space, two Gâteaux differentiable functionals, . We say that functional satisfies the PalaisSmale condition cut off upper at r (in short condition) if any sequence , such that
has a convergent subsequence.
When , the previous definition is the same as the classical definition of the PalaisSmale condition, while if , such a condition is more general than the classical one. We refer to [25] for more details.
Theorem 2.1 (see [25], Theorem 7.4)
LetXbe a real Banach space, and letbe two continuously Gâteaux differentiable functionals such that Φ is bounded from below. For every, let us put
and
(a) Ifand for each, the functionalsatisfies thecondition for all, then for each, the following alternative holds: either
or
(a_{2}) there exists a sequenceof critical points (local minima) ofsuch that.
(b) Ifand for each, the functionalsatisfies thecondition for all, then for each, the following alternative holds: either
(b_{1}) there exists a global minimum of Φ which is a local minimum of
or
(b_{2}) there exists a sequence of pairwise distinct critical points (local minima) ofsuch that.
We recall that Theorem 2.1 improves [[26], Theorem 2.5] since no assumptions with respect to weak topology of X are made. In particular, the set is not involved in the definition of φ and the sequential weak lower semicontinuity of is not required.
3 Main results
In this section, we present our main results. Put
Moreover, let
Our first result is as follows.
Theorem 3.1Assume that
Then, for everyand for every continuous function, , whose potential, , satisfies
such that for every, problem () has an unbounded sequence of weak solutions.
Proof First, we observe that owing to (a_{2}) the interval Λ is nonempty. Moreover, for each and taking into account that , one has . Now, fix λ and μ as in the conclusion. Our aim is to apply Theorem 2.1. For this end, take and Φ, Ψ as in (3).
We divide our proof into three steps in order to show Theorem 3.1. First, we prove that satisfies the condition for all . So, fix and let be a sequence such that is bounded, and for all . From , taking into account that Φ is coercive, is bounded in X. Since the embedding of X in is compact (see, for instance, [[27], Theorem 8.8]) and X is reflexive, up to a subsequence, is uniformly convergent to , and is weakly convergent to in X. The uniform convergence of , taking also into account Lebesgue’s theorem, ensures that
that is,
Now, from , there is a sequence , with , such that
for all with and for all . Setting = , one has
for all . Moreover, having in mind that , one has
that is,
From (5) and (6) one has
and owing to (4), one has
Hence, [[27], Proposition III.30] ensures that strongly converges to and our claim is proved.
Second, we wish to prove that
Let be a sequence of positive numbers such that and
Put for all . By Lemma 1, for all , one has
for all such that . Hence, one has
So, from assumptions (a_{2}) and (i_{2}),
that is, . The previous inequality assures that conclusion (a) of Theorem 2.1 can be used, for which either has a global minimum or there exists a sequence of solutions of problem () such that .
The final step is to verify that the functional has no global minimum. From , and taking into account that , there is such that
So, there exists a sequence of positive numbers such that and
It follows that there is such that for all , one has
Now, consider a function defined by setting
Clearly, one has
Moreover, bearing in mind (a_{1}) and (i_{1}),
Putting together (7) and (8), we get that the functional is unbounded from below and so it has no global minimum.
Therefore, Theorem 2.1 assures that there is a sequence of critical points of such that and, taking into account the considerations made in Section 2, the theorem is completely proved. □
Remark 3.1 Assume that . Clearly, condition (a_{1}) holds, and condition (a_{2}) assumes the following simpler form:
In particular, if and , then () holds and problem () has an unbounded sequence of weak solutions in X for every pair .
Moreover, under the assumption , Theorem 3.1 guarantees the existence of infinitely many solutions to problem () for every .
As an example, we point out below a special case of Theorem 3.1.
Corollary 3.1Letbe a continuous function, putfor every, and let. Assume that
Then, for each, and for each continuous functionsuch that, , the problem
admits infinitely many pairwise distinct classical solutions.
Replacing the condition at infinity of the potential F by a similar one at zero, and arguing as in the proof of Theorem 3.1 but using conclusion (b) of Theorem 2.1 instead of (a), one establishes the following result. Put
Theorem 3.2Assume that
Then, for everyand for every continuous function, , whose potential, , satisfies
such that for every, problem () has a sequence of nonzero weak solutions, which strongly converges to 0.
Proof We take X, Φ and Ψ as in the proof of Theorem 3.1. Fix , let be a function that satisfies assumptions (i_{1}) and (j_{2}) and take . Arguing as in the proof of Theorem 3.1, one has . Now, arguing again as in the proof of Theorem 3.1, there is a sequence of positive numbers such that and for all and for some . By choosing as in the proof of Theorem 3.1, the sequence strongly converges to 0 in X and for each . Therefore, taking into account that , 0 is not a local minimum of . The part (b) of Theorem 2.1 ensures that there exists a sequence in X of critical points of such that and the proof is complete. □
Let be a primitive of , an Carathéodory function and put
Moreover, let
In virtue of Theorems 3.1 and 3.2, we obtain the following results for problem ().
Theorem 3.3Assume that
Then, for everyand for every continuous function, , whose potential, , satisfies
such that for each, problem () has an unbounded sequence of weak solutions.
Proof As seen in Section 2, we put , and , . Clearly, one has , , , , . Hence, from Theorem 3.1 the conclusion is achieved. □
Remark 3.2 Theorem 1.1 in Introduction is an immediate consequence of Theorem 3.3. In fact, it is enough to observe that (c_{1}) is verified and one has and , for which . Moreover, from , one has and the conclusion is achieved.
Replacing the condition at infinity of the potential G by a similar one at zero, one establishes the following result. Put
Theorem 3.4Assume
Then, for everyand for every continuous function, , whose potential, , satisfies
such that for each, problem () has a sequence of nonzero weak solutions, which strongly converges to 0.
Proof The conclusion follows from Theorem 3.2 by arguing as in the proof of Theorem 3.3. □
Remark 3.3 We point out that in Theorem 3.3 (as in Theorem 3.4) the assumption can be deleted provided that we assume the constant and the interval .
Finally, we observe that the existence of infinitely many solutions to problem () can be obtained from Theorem 3.3 and Theorem 3.4 even under small perturbations of the nonlinearity. As an example, we point out the following consequence of Theorem 3.3.
Corollary 3.2Letbe anCarathéodory function satisfying (c_{1}) and (c_{2}) of Theorem 3.3.
Then, for every, for every nonnegativeCarathéodory function, whose potentialsatisfies
and for every continuous function, , whose potential, , satisfies (i_{1}) and (i_{2}) of Theorem 3.3, there existand, where
such that for alland for all, the problem
has an unbounded sequence of weak solutions.
Proof It is enough to apply Theorem 3.3 to the following function:
where is fixed in and is fixed in Λ. In fact, one has
and
for which , that is, . Moreover, from (9) one has and from (10) . Hence, and Theorem 3.3 ensures the conclusion. □
4 Applications
In many papers [13,20,22,28] and [23], the authors obtain the existence of infinitely many solutions for problem () while the impulsive term is supposed to be odd. The next examples provide problems that admit infinitely many solutions for which those other results cannot be applied.
Example 4.1 Consider the following boundary value problem:
where is the function defined as follows:
It is easy to see that conditions (a_{1}), (a_{2}), (i_{1}) and (i_{2}) of Theorem 3.1 hold. In particular, and
Then, for each and for every , problem (11) has an unbounded sequence of solutions in X.
Now, we give an application of Theorem 3.4.
Example 4.2 Consider the Dirichlet problem
where is the function defined as follows:
By a simple calculation, we get and
Then, from Theorem 3.4, for each and for every , problem (12) admits a sequence of pairwise distinct classical solutions strongly converging at 0. We observe that, in this case, as direct computations show, also zero is a solution of the problem.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have contributed equally to this research work. All authors read and approved the final manuscript.
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