Eigenvalue problem of an impulsive di ﬀ erential equation governed by the one-dimensional p-Laplacian operator

In this paper we study a nonlinear boundary eigenvalue problem governed by the one-dimensional p-Laplacian operator with impulse, we give some properties of the ﬁ rst eigenvalue λ 1 and we prove the existence of eigenvalues sequence { λ n } n ∈ N ∗ by using the Lusternik-Schnirelman principle, as well as by the characterization of the sequence of eigenvalues, we discuss the strict monotonicity of the ﬁ rst eigenvalue and we prove that the eigenfunction corresponding to second eigenvalue λ 2 changes sign only once on [0 , 1] .


Introduction
Impulsive differential equations describe several phenomena in many fields. The idea of studying processes which can change state suddenly is natural and appears to be a good model for some applications in real world. For example, one of these applications cited in [1], is a pharmacokinetic model, involving first-order processes for drug release, this last is known as Kruger-Thiemer model. In this model the authors assume that the drug taken by a patient is nearly digested. Thus the time of the very process of absorbing is abandoned. Such a model of the process leads to impulsive differential equations, with impulses which take place when the drug is taken. Impulsive differential equation also study models in epidemiology [2], chemistry [3], economics [4], optimal control theory [5], nonlinear mechanics [6]. For the general theory of impulsive differential equations, we refer the reader to the references [7] and [8]. Some approaches have been used to study such problems in the literature. These approaches include the degree theory [9], the techniques of upper and lower solutions [10], and the fixed point theorems [11]. On the other hand, many researchers have used a variational method to the existence and multiplicity of solutions with impulsive effect (see [12], [13], [14], [15], [16] and [17]). The spectrum of the equations that involve the one-dimensional p-Laplacian operator with the different boundary condition, has been studied by several authors, for literature we quote here some works [18], [19], [20] and [21].
Several results have been obtained on this type of eigenvalue problems of differential equations, see, for instance, [22], [23] and [24]. A principal result is that (1.5) − (1.6) has a non-decreasing sequence of eigenvalues {λ n } n∈N * which tend to ∞ as n −→ ∞. More precisely, in [24] the authors studied a eigenvalue problems of differential equations with impulsive effects which is given as below by They proved that the eigenvalue problem (1.7)−(1.9) has an unbounded sequence of eigenvalues {λ n } n∈N * . In the present paper, we will study the general case of problem (1.1) − (1.3) when p 6 = 2. That is, we consider the following nonlinear eigenvalue problem with impulsive effects is not a solution of (1.10) − (1.12), because this latter is not continuous throughout the interval J = [0, 1].
Motivated by [21], we will establish some new properties for the eigenvalue problem (1.10)−(1.12). This paper is organized as follows. In Section 2, we present some preliminary results. In Section 3, the main results of this paper will be presented.

Proof.
Let Then, for k = 1, the problem IVP(u 0 , u 0 ) is lied as follows We reformulate the problem (2.2) to a Cauchy problem through which we distinguish two cases. Firstly, when t ∈ [0, t 1 [, we have the following Cauchy problem which is locally Lipschitzian with respect to the second variable X 1 in a neighborhood V 1 which contains X 1 (0). In fact, we provide the space R 2 by the Euclidean norm and let W (t) = (y(t), u(t)) T , Z(t) = (z(t), v(t)) T ∈ V 1 , such that we have , as well that K p and K q are the Lipschitz constants of Φ p and Φ q respectively.
Then by Cauchy-Lipschitz theorem, we have the existence and uniqueness of solution in a neighborhood which contains 0.
Finally, when t ∈ [t 1 , 1], we have a new Cauchy problem that is given as follows Similarly to the above, by applying the Cauchy-Lipschitz theorem we show the existence and uniqueness of solution in a neighborhood which contains t 1 . Thus, for k = 1, the problem (2.2) has a unique solution as follows The other cases k = 2, 3, . . . , i are treated in the same way as in the case k = 1. Consequently, we have the existence and uniqueness of solution for the problem (2.2), which is given as follows 2 Lemma 2.2. Let y and z, the two solutions of (1.10), (1.11) with the same value λ. Then, there exists a constant c such that y = cz or z = cy.
In addition, g 0 (ξ) = 0 if and only if ξ = 1 2 , then g admits a minimum Lemma 2.4. For the space W 1,p 0 ([0, 1]), the norms kuk 1,p and kuk are equivalent, and there is c 1] s(t). We have On the other hand, we have For u ∈ W 1,p 0 ([0, 1]), it follows from the mean value theorem that Hence, for t ∈ [0, 1], using Hölder's inequality with 1 p which completes the proof.
(2) For any u ∈ M , any sequence {u n } in M such that u n u weakly in Then, E is bounded from below on M and is attained its infimum in M .

Main results
Let In what follows, we assume that for a k ∈ R.
Let {u n } n∈N be a minimizing sequence on Ω 1 such that F (u n ) −→ λ 1 as n −→ +∞. Hence, the above inequality implies that there exists c 0 > 0 such that F (u n ) ≥ c 0 ku n k p . Therefore, the sequence {u n } n∈N is bounded.

Proof.
Let u 1 ∈ Ω 1 be an eigenfunction that corresponds to the first eigenvalue λ 1 . Then, u 1 satisfies the following equations The eigenfunction y = |u 1 | satisfies the above equations. Hence, y is also a solution of (1.

The Ljusternik-Schnirelman principle
Let W 1,p 0 ([0, 1]) be a real reflexive Banach space and F, G be two functionals on W 1,p 0 ([0, 1]). Consider the following eigenvalue problem (H 4 ) The set S F is bounded and u 6 = 0 implies It is known that (u, µ) solves (3.10) if and only if u is a critical point of G with respect to S F . For any positive integer n, denoted by C n the class of all compact, symmetric subsets K of S F such that G(u) > 0 on K and γ(K) ≥ n, where γ(K) denotes the genus of K, i.e., γ(K) = inf{k ∈ N : ∃h : K −→ R k \{0} such that h is continuous and odd }. Let  (2) If χ = ∞, (3.10) has infinitely many pairs (−u, +u) of eigenfunctions which correspond to nonzero eigenvalues.

Proof.
Let's show that F 0 and G 0 are continuous. Denote by k.k * the norm of the dual space W 1,p 0 ([0, 1]). Let {u n } n∈N be a sequence in By Hölder's inequality with the compact embedding of W 1,p 0 ([0, 1]) in L p ([0, 1]) and Lemma 2.4, we have dividing the last inequality by kvk, we obtain Which implies the continuity of F 0 . The proof of the continuity of G 0 is the same as F 0 .
γ is the genus function defined below. We also define the set B n = {C ∈ C n : C ⊂ Ω}. Then, (3.19) can be rewritten as

Proof.
Let m 1 and m 2 be such that m 1 (t) < m 2 (t), for all t ∈ J. We know previously that the first eigenfunction u 1 ∈ S F corresponding to λ 1 (m, J) has no zero in J, i.e u 1 (t) 6 = 0 for all t ∈ J.
According to (3.20), we have To prove the second inequality, we consider I a sub interval of J and m /I a weight defined on I. Let u 1 ∈ S F be a positive eigenfunction associated to λ 1 (m /I , I), and denote by u 1 the extension by zero on J. Then, we have Therefore, the last strict inequality holds from the fact that u 1 vanishes in J\I. Thus, the latter cannot be a eigenfunction which correspond to the first eigenvalue λ 1 (m, J).

Proof.
We start by showing that un has a finite number of nodal domains. Assume that there exists a sequence J k , k ≥ 1, of nodal domains, I l ∩ I l 0 = ∅ for l 6 = l 0 . Combining Lemma 3.2 with Proposition 3.7, we get which is a contradiction with the fact that J is bounded.