Quasilinear boundary value problem with impulses: variational approach to resonance problem
© Drábek and Langerová; licensee Springer. 2014
Received: 5 December 2013
Accepted: 7 March 2014
Published: 24 March 2014
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.
Keywordsquasilinear impulsive differential equations Landesman-Lazer condition variational methods critical point theory linking theorem
where is a spectral parameter and , , is a given right-hand side.
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 .
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)
Then (1), (2) has a solution for arbitrary .
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., ).
Our main result is the following.
Theorem 2 (Resonance case)
Let for some in (1). Let the nonlinear bounded impulse functions , , and the right-hand side satisfy either (5) or (6). Then (1), (2) has a solution.
The result from Theorem 2 is illustrated in the following special example.
2 Functional framework
We say that u is the classical solution of (1), (2) if the following conditions are fulfilled:
, , is absolutely continuous in , ;
the equation in (1) holds a.e. in and ;
one-sided limits , exist finite and (2) holds.
holds for any function .
and hence also (2) follows. Similarly, we show that every classical solution is a weak solution at the same time.
is -homogeneous, odd, continuously invertible, and for any .
is -homogeneous, odd and compact.
is bounded and compact.
By the linearity of , is a fixed element.
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. ).
for any continuous map such that , there holds .
(See [, Def. 8.1, p.116].)
Theorem 3 (Linking theorem)
Then is a critical value of ℱ.
(See [, Thm. 8.4, p.118].)
The purpose of the following series of lemmas is to show that the hypotheses of Theorem 3 are satisfied provided that either (5) or (6) holds. From now on we assume that (for some ) in (1).
Lemma 2 If either (5) or (6) is satisfied, then ℱ satisfies the Palais-Smale condition.
By the boundedness of we know that . We also have . By the compactness of we get in . Thus in X by Lemma 1(A). It follows that .
which contradicts (5) or (6). Hence is bounded.
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.
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.
if or if ;
is strictly decreasing in t if and ;
for with equality if and only if for some .
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.
Lemma 4 If (6) is satisfied, then there exist and such that for any and .
Since is compact, we may assume, without loss of generality, that in for some .
for . However, this contradicts (17).
if is a zero point of for some . The case is proved similarly using the second inequality in (6). □
for all , . In particular, for all and (18) is proved. □
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.
Next we show that the sets and satisfy the hypotheses of Theorem 3 if (5) is satisfied.
The principal difference consists in the fact that, in contrast with , the set is not compact. That is why one more technical lemma is needed.
for . (Here is the ball in X centered at with radius .)
is a locally Lipschitz continuous symmetric pseudo-gradient vector field associated with E on and is a smooth function such that for u satisfying and for u satisfying or .
with . □
The following lemma is a counterpart of Lemma 4 in the case of condition (5).
Lemma 7 If (5) is satisfied, then there exist and such that for any and .
by the second inequality in (5). This contradicts (20) again. The case as is proved similarly but using the first inequality in (5). □
and (21) is proved. □
It follows that the sets and satisfy the hypotheses of Theorem 3 if (5) is satisfied. The proof of Theorem 2 is thus completed.
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.
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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