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
We establish some multiplicity results for problem (
Theorem 1.1Let
Then there is
admits infinitely many pairwise distinct classical solutions.
We explicitly observe that in Theorem 1.1 impulsive effects
2 Preliminaries
By a classical solution of (
that satisfies the equation in (
We consider the following slightly different form of problem (
where
It is easy to see that, by choosing
the solutions of (
Let us introduce some notations. In the Sobolev space
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])
Let
where
Here, and in the sequel,
(a)
(b)
(c) for every
for almost every
Definition 1 A function
for any
Lemma 2 ([[24], Lemma 2.1 ])
Now, we define the functionals
for each
and
So, arguing in a standard way, it is possible to prove that the critical points of
the functional
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,
(α)
(β)
(γ)
has a convergent subsequence.
When
Theorem 2.1 (see [25], Theorem 7.4)
LetXbe a real Banach space, and let
and
(a) If
(a_{1})
or
(a_{2}) there exists a sequence
(b) If
(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) of
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
3 Main results
In this section, we present our main results. Put
Moreover, let
Our first result is as follows.
Theorem 3.1Assume that
(a_{1})
(a_{2})
Then, for every
(i_{1})
(i_{2})
there exists
such that for every
Proof First, we observe that owing to (a_{2}) the interval Λ is nonempty. Moreover, for each
We divide our proof into three steps in order to show Theorem 3.1. First, we prove
that
that is,
Now, from
for all
for all
that is,
From (5) and (6) one has
and owing to (4), one has
Hence, [[27], Proposition III.30] ensures that
Second, we wish to prove that
Let
Put
for all
So, from assumptions (a_{2}) and (i_{2}),
Assumption
that is,
The final step is to verify that the functional
So, there exists a sequence of positive numbers
It follows that there is
Now, consider a function
Clearly, one has
Moreover, bearing in mind (a_{1}) and (i_{1}),
Putting together (7) and (8), we get that the functional
Therefore, Theorem 2.1 assures that there is a sequence
Remark 3.1 Assume that
(
In particular, if
Moreover, under the assumption
As an example, we point out below a special case of Theorem 3.1.
Corollary 3.1Let
Then, for each
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
(a_{1})
(b_{2})
Then, for every
(i_{1})
(j_{2})
there exists
such that for every
Proof We take X, Φ and Ψ as in the proof of Theorem 3.1. Fix
Let
Moreover, let
In virtue of Theorems 3.1 and 3.2, we obtain the following results for problem (
Theorem 3.3Assume that
(c_{1})
(c_{2})
Then, for every
(i_{1})
(i_{2})
there exists
such that for each
Proof As seen in Section 2, we put
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
Replacing the condition at infinity of the potential G by a similar one at zero, one establishes the following result. Put
Theorem 3.4Assume
(c_{1})
(
Then, for every
(i_{1})
(j_{2})
there exists
such that for each
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
Finally, we observe that the existence of infinitely many solutions to problem (
Corollary 3.2Let
Then, for every
and for every continuous function
such that for all
has an unbounded sequence of weak solutions.
Proof It is enough to apply Theorem 3.3 to the following function:
where
and
for which
4 Applications
In many papers [13,20,22,28] and [23], the authors obtain the existence of infinitely many solutions for problem (
Example 4.1 Consider the following boundary value problem:
where
It is easy to see that conditions (a_{1}), (a_{2}), (i_{1}) and (i_{2}) of Theorem 3.1 hold. In particular,
Then, for each
Now, we give an application of Theorem 3.4.
Example 4.2 Consider the Dirichlet problem
where
By a simple calculation, we get
Then, from Theorem 3.4, for each
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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