This paper deals with the resonance problem for the one-dimensional p-Laplacian with homogeneous Dirichlet boundary conditions and with nonlinear impulses in the derivative of the solution at prescribed points. The sufficient condition of Landesman-Lazer type is presented and the existence of at least one solution is proved. The proof is variational and relies on the linking theorem.
MSC: 34A37, 34B37, 34F15, 49K35.
Keywords:quasilinear impulsive differential equations; Landesman-Lazer condition; variational methods; critical point theory; linking theorem
For the sake of brevity, in further text we use the following notation:
For this problem is considered in  where the necessary and sufficient condition for the existence of a solution of (1) and (2) is given. In fact, in the so-called resonance case, we introduce necessary and sufficient conditions of Landesman-Lazer type in terms of the impulse functions , , and the right-hand side f. They generalize the Fredholm alternative for linear problem (1) with .
In this paper we focus on a quasilinear equation with and look just for sufficient conditions. We point out that there are principal differences between the linear case () and the nonlinear case (). In the linear case, we could benefit from the Hilbert structure of an abstract formulation of the problem. It could be treated using the topological degree as a nonlinear compact perturbation of a linear operator. However, in the nonlinear case, completely different approach must be chosen in the resonance case. Our variational proof relies on the linking theorem (see ), but we have to work in a Banach space since the Hilbert structure is not suitable for the case .
It is known that the eigenvalues of
are simple and form an unbounded increasing sequence whose eigenspaces are spanned by functions such that has evenly spaced zeros in , , and . The reader is invited to see [, p.388], [, p.780] or [, pp.272-275] for further details. See also Example 1 below for more explicit form of and .
Let , , in (1). This is the nonresonance case. Then, for any , there exists at least one solution of (1). In the case , this solution is unique. In the case , the uniqueness holds if , but it may fail for certain right-hand sides if . See, e.g.,  (for ) and  (for ).
The same argument as that used for in [, Section 3] for the nonresonance case yields the following existence result for the quasilinear impulsive problem (1), (2).
Theorem 1 (Nonresonance case)
Variational approach to impulsive differential equations of the type (1), (2) with was used, e.g., in paper . The authors apply the mountain pass theorem to prove the existence of a solution for . Our Theorem 1 thus generalizes [, Theorem 5.2] in two directions. Firstly, it allows also (, ) and, secondly, it deals with quasilinear equations (), too.
is the sufficient condition for solvability of (1), but it is not necessary if . Moreover, if but f is ‘close enough’ to , problem (1) has at least two distinct solutions. The reader is referred to  or  for more details. It appears that the situation is even more complicated for , (see, e.g., ).
In the presence of nonlinear impulses which have certain asymptotic properties (to be made precise below), we show that the fact might still be the sufficient condition for the existence of a solution to (1) (with ) and (2). For this purpose we need some notation. Let denote evenly spaced zeros of , let and denote the union of intervals where or , respectively. We arrange , , into three sequences: , , ; , , ; , . Obviously, we have and . Assume that , i.e., . The impulse condition (2) can be written in an equivalent form
Our main result is the following.
Theorem 2 (Resonance case)
The result from Theorem 2 is illustrated in the following special example.
Example 1 It follows from the first integral associated with the equation in (3) that the eigenvalues and the eigenfunctions of (3) have the form
where and , , , , , , see [, p.388]. Let us consider in (1) and , , , , , in (2). Since , , condition (6) reads as follows:
2 Functional framework
We say that u is the classical solution of (1), (2) if the following conditions are fulfilled:
Integration by parts and the fundamental lemma in calculus of variations (see [, Lemma 7.1.9]) yields that every weak solution of (1), (2) is also a classical solution and vice versa. Indeed, let u be a weak solution of (1), (2), (the space of smooth functions with a compact support in , ), elsewhere in , then
and hence also (2) follows. Similarly, we show that every classical solution is a weak solution at the same time.
Proof See [, Lemma 10.3, p.120]. □
As mentioned already above, in the nonresonance case (, ), we can use the Leray-Schauder degree argument and prove the existence of a solution of the equation (9) exactly as in [, proof of Thm. 1]. Note that the -subhomogeneous condition on is used here instead of the sublinear condition imposed on in  and the proof of Theorem 1 follows the same lines. For this reason we skip it and concentrate on the resonance case ( for some ) in the next section.
3 Resonance problem, variational approach
We use the following definition of linked sets and the linking theorem (cf.).
Definition 1 Let ℰ be a closed subset of X and let Q be a submanifold of X with relative boundary ∂Q. We say that ℰ and ∂Q link if
(See [, Def. 8.1, p.116].)
Theorem 3 (Linking theorem)
(See [, Thm. 8.4, p.118].)
Lemma 2If either (5) or (6) is satisfied, then ℱ satisfies the Palais-Smale condition.
Proof Suppose that such that and in . We must show that has a subsequence that converges in X. We prove first that is a bounded sequence. We proceed via contradiction and suppose that and consider . Without loss of generality, we can assume that there is such that (weakly) in X (X is a reflexive Banach space). Since
and the Cauchy-Schwarz inequality implies
Recall that X embeds compactly in , so, without loss of generality, we assume that , , as . Hence, for , which implies as well as as by an application of the l’Hospital rule to . Notice that by the boundedness of we have
By compactness there is a subsequence such that and converge in (see Lemma 1(B), (J)). Since by our assumption, we also have that converges in . Finally, converges in X by Lemma 1(A). The proof is finished. □
With the Palais-Smale condition in hands, we can turn our attention to the geometry of the functional ℱ. To this end we have to find suitable sets which link in the sense of Definition 1. Actually, we use the sets constructed in  and explain that they fit with the hypotheses of Theorem 3 if either (5) or (6) is satisfied.
Consider the even functional
and the manifold
It is proved in [, Section 3] that is a sequence of eigenvalues of homogeneous problem (3). It then follows from the results in  that this sequence exhausts the set of all eigenvalues of (3) with the properties described in Section 1.
Notice that the second equality holds thanks to the fact
a super-level set, and
It is proved in [, Lemma 7] that the couple and satisfies condition (ii) from Definition 1. It is also proved in [, Lemma 8] that the couple and satisfies the same condition. To show that also other hypotheses of Theorem 3 are satisfied, we need some technical lemmas.
Now we can finish the proof of Theorem 2 under assumption (6). Indeed, it follows from (18) that and thus the hypotheses of Theorem 3 hold with and . It then follows that ℱ has a critical point and hence (1), (2) has a solution.
Hence for . Since E satisfies the Palais-Smale condition on (see [, Lemma 2]), there exists such that for all . Then
The following lemma is a counterpart of Lemma 4 in the case of condition (5).
and (21) is proved. □
Final remark Reviewers of our manuscript suggested to include some recent references on impulsive problems. Variational approach to impulsive problems can be found, e.g., in [17-21]. The last reference deals with the p-Laplacian with the variable exponent . Singular impulsive problems are treated in [22-24]. Impulsive problems are still ‘hot topic’ attracting the attention of many mathematicians and the bibliography on that topic is vast.
The authors declare that they have no competing interests.
Both authors read and approved the final manuscript.
This research was supported by Grant 13-00863S of the Grant Agency of Czech Republic and by the European Regional Development Fund (ERDF), project ‘NTIS - New Technologies for the Information Society’, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.
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