Variational approach to non-instantaneous impulsive di ﬀ erential equations with p -Laplacian operator

: In this paper, we consider the existence, multiplicity and nonexistence of solutions for a class of p -Laplacian di ﬀ erential equations with non-instantaneous impulses. By using variational methods and critical point theory, we obtain that the impulsive problem has at least one nontrivial solution, at least two nontrivial solutions and no nontrivial solution.

In recent years, variational methods and critical point theory have been widely used to study the existence and multiplicity of solutions for impulsive differential equations which possess variational structures under certain boundary conditions. In this field of research, the pioneering studies were initiated by Tian-Ge [29] and Nieto-O'Regan [21]. Since this time, many scholars investigated different types of impulsive differential equations, such as second-order, fourth-order and fractional order impulsive differential equations, by means of variational approach and critical point theory. For some general and recent works, we refer the interested reader to [11,12,15,20,22,23,28,32,33,35] and some of references for additional details.
On the other hand, the p-Laplacian operator appears in non-Newtonian fluid flows, turbulent filtration in porous media, some reaction-diffusion and many other application areas, so it has deep background in physics [3,10,17]. In 1983, Leibenson [17] considered the one-dimensional turbulent flow of a polytropic gas in a porous medium and introduced the p-Laplacian equation as follows: (Φ p (u (t))) = f (t, u(t), u (t)), (1.2) where Φ p (x) = |x| p−2 x, p > 1. Obviously, when p = 2, the p-Laplacian equations reduce to the secondorder differential equations. Many important results relative to (1.2) have been achieved over the past few decades, see for instance [4,6,9,31]. In 2008, Tian-Ge [29] first investigated a class of p-Laplacian boundary value problem with impulsive effects via variational methods and obtained it has at least two positive solutions. Since then, the impulsive differential equations with p-Laplacian operators have received extensive attention. For example, in [5,13,14,19,25,26,30], authors studied different types of p-Laplacian impulsive differential equations by applying variational methods and critical point theory.
Although the above works about impulsive differential equations have achieved many important research results, they focused primarily on differential equations with instantaneous impulses, which can't describe all the phenomena in real life, such as earthquakes and tsunamis. Thus, the study of noninstantaneous impulsive differential equations has attracted widespread attention in recent years. In 2013, Hernández-O'Regan [16] were inspired by a simplified situation concerning the hemodynamical equilibrium of a person and introduced the non-instantaneous impulses, which start abruptly at some points and remain active on a finite time interval. Obviously, non-instantaneous impulsive differential equations are a natural generalization of impulsive differential equations. From then on, the existence of solutions for non-instantaneous impulsive differential equations have been investigated by some approaches, such as fixed point theory, theory of analytic semigroup and variational methods, see for instance [1,2,7,16,24,36]. In [1], Bai-Nieto studied the following linear differential equations with non-instantaneous impulses and obtained the existence and uniqueness of weak solutions.
where 0 = s 0 < t 1 < s 1 < t 2 < s 2 < ... < t N < s N < t N+1 = T , the impulses start abruptly at the points t i and keeps the derivative constants on finite interval (t i , s i ], σ i ∈ L 2 ((s i , t i+1 ), R) and α i are given constants. Here, u (s ± i ) = lim s→s ± u (s). On the basis of [1], Bai-Nieto-Wang [2] considered the following nonlinear differential equations with non-instantaneous impulses via variational methods and critical point theory. They obtained the problem has at least two distinct nontrivial weak solutions.
where 0 = s 0 < t 1 < s 1 < t 2 < s 2 < ... < t N < s N < t N+1 = T , the impulses start abruptly at the points t i and keeps the derivative constants on finite interval (t i , s i ]. Here, u (s ± i ) = lim s→s ± u (s) and α i are given constants.
In [36], Zhao-Luo-Chen considered the following fractional differential equations with noninstantaneous impulses. They proved that the problem has at least one nontrivial weak solution, at least two nontrivial weak solutions and does not admit any nontrivial solution by using critical point theory and variational methods.
where 1 2 < α ≤ 1, 0 D γ t and t D γ T denote the left and right Riemann-Liouville fractional derivatives of order γ, respectively. c 0 D α t is the left Caputo fractional derivative of order α, 0 = s 0 < t 1 < s 1 < t 2 < s 2 < ... < s N < t N+1 = T , β i are given constants, and the impulsive jump starts abruptly at the fixed points t i and keeps the derivative constants on finite interval (t i , , and the nonlinear functions D x F i (t, x) are the derivatives of F i (t, x) with respect to x for every i = 0, 1, 2, ..., N.
Motivated by the above research, our aim of this paper is to study a class of non-instantaneous impulsive differential equations with p-Laplacian operator. As far as we know, there is no paper considered the problem (1.1) and the existence, multiplicity and nonexistence of solutions for it are obtained via variational methods and critical point theory. So we generalize the existing results in [1,2,36].
Throughout this paper, we need the following assumptions.
(H1) F i (t, x) is measurable in t for every x ∈ R and continuously differentiable in x for a.e. t ∈ (s i , t i+1 ], and there exist functions for each x ∈ R and a.e. in t ∈ (s i , t i+1 ]. (H2) There exist constant γ ∈ [0, p) and the for a.e. t ∈ (s i , t i+1 ] and x ∈ R.
(H3) F i (t, 0) = 0 for a.e. t ∈ (s i , t i+1 ] and there exist constants µ i > p and M > 0 such that 0 and C is defined in Lemma 2.10. Our main results are as follows.  ii The conditions (H3) and (H4) ensure ϕ(u) has mountain pass geometric structure. So we can obtain the problem (1.1) has at least one nontrivial weak solution by using mountain pass theorem. Furthermore, we can get another nontrivial weak solution of the problem (1.1) by applying Lemma 2.7. Therefore, the Theorem 1.2 is proved. Similarly, we can prove Theorem 1.3 by mean of the same methods.
The paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we will prove our main results.

Preliminaries
In this section, we first introduce some definitions, lemmas and theorems, which are used further in this paper.
is coercive, then ϕ has a bounded minimizing sequence.
[18] Let X be a reflexive Banach space and let ϕ : X → (−∞, +∞] be weakly lower semi-continuous on X. If ϕ has a bounded minimizing sequence, then ϕ has a minimum on X. Definition 2.6. ( [18], (PS) condition) Let X be a real reflexive Banach space. For any sequence (i) X is a real reflexive Banach space; (ii) M is bounded and weak sequentially closed; (iii) ϕ is sequentially weakly lower semi-continuous on M.
Next, we recall the well-known Poincaré inequality where λ 1 is the first eigenvalues of the following problem Moreover, it has been shown in [8] that all eigenvalues of the problem (2.1) are given by the sequence of positive numbers Obviously, X is a reflexive Banach space. We also define the norms in For each v ∈ X, we have On the other hand, (2.4) Definition 2.9. A function u ∈ X is said to be a weak solution of problem (1.1) if (2.4) holds for any v ∈ X.
We define the following functional on X For u and v fixed in X and λ ∈ [−1, 1], by Lemma 2.10, we have and It follows from condition (H1), (2.7) and mean value theorem that for some θ ∈ (0, 1). By Lebesgue's dominated convergence theorem, we obtain that φ i has at every point u a directional derivative Thus, φ i ∈ X * (The space X * is the topological dual of X). Suppose u k u in X, then u k → u on [0, T ] [18, Proposition 1.2]. Furthermore, by (2.8), we obtain Hence, φ i (u) is continuous from X to X * , ϕ ∈ C 1 (X, R) and

Main results
Lemma 3.1. The functional ϕ : X → R is weakly lower semi-continuous.
Proof of Theorem 1.2. We need four steps to complete the proof.
Let {u k } ⊂ X such that ϕ(u k ) is a bounded sequence and lim From (H3), we have which implies that {u k } is bounded in X. Since X is a reflexive Banach space, going to a subsequence if necessary, we may assume that u k u in X, then u k → u in L p [0, T ] and u k → u a.e. t ∈ [0, T ]. Since lim k→∞ ϕ (u k ) = 0 and {u k } converges weakly to some u, one has (3.6) From (2.9), we have (3.7) In view of assumption (H1) and we obtain that the second term on the right hand of (3.7) converges to 0 as k → ∞. By [27, Eq (2. 2)], there exist c p , d p > 0 such that If p ≥ 2, it follows from (3.6) and (3.7) that u k − u X → 0 as k → ∞. If 1 < p < 2, by Hölder's inequality, one has (3.8) It follows from 1 < p < 2 and (3.8) that (3.9) In view of (3.6)-(3.9), one has u k − u X → 0 as k → ∞. Thus, {u k } converges strongly to u in X.
Since Ω ρ is closed-convex set, then Ω ρ is sequentially weakly closed. Moreover, ϕ(u) is continuous differential and sequentially weakly lower semi-continuous on X as the sum of a convex function and of a weakly continuous one. Therefore, it follows from Lemma 2.7 that there exists a u * ∈ Ω ρ such that ϕ(u * ) = min Ω ρ ϕ(u).
Proof of Theorem 1.3. According to condition (H5) and (3.10), for any > 0, there exists a η := A * n i=0 (t i+1 − s i ) such that the functional ϕ(u) ≥ η > 0. By the similar proof of Theorem 1.2, we can obtain Theorem 1.3 holds, and we omit the rest of proof of it.
Proof of Theorem 1.4. Let u be a nontrivial solution of problem (1.1), then we obtain By virtue of condition (H6), (2.6), we have Furthermore, in view of conditions (H1) and (H7), we obtain which is a contradiction. Hence, problem (1.1) has no nontrivial solution.