Multiplicity of Solutions for Dirichlet Boundary Conditions of Second-order Quasilinear Equations with Impulsive Effects

This paper deals with the multiplicity of solutions for Dirichlet boundary conditions of second-order quasilinear equations with impulsive effects. By using critical point theory, a new result is obtained. An example is given to illustrate the main result.

It is generally known that critical point theory is a classical method to deal with the existence and multiplicity of solutions for differential equations (see [3,7,12,16,21,26,30]).Then a natural question is asked: Can we consider the multiplicity of solutions for secondorder quasilinear equations with impulsive effects which are produced by the quasilinear term (|u| 2 ) u and u by using critical point theory?
Impulsive differential equations can be used to describe many evolution processes (see [5,10,11,14,17,27]).Some classical methods and theorems such as fixed point theorems, the method of lower and upper solutions and coincidence degree theory have been widely used to investigate impulsive differential equations (see [1,8,13,15,20]).Recently, critical point theory has been proved to be an effective tool to investigate boundary value problems for impulsive differential equations.Many valuable results have been obtained by some scholars (see [6,18,24,25,28,29]).
In [18], Nieto and O'Regan studied the linear Dirichlet problem with impulses and the nonlinear Dirichlet problem with impulses and got some results by using critical point theory.
In [29], Zhou and Li investigated the nonlinear Dirichlet problem with impulses and obtained the existence of infinitely many solutions by employing the Symmetric Mountain Pass Theorem.However, there are few articles which considered the multiplicity of standing wave solutions for the impulsive Dirichlet boundary value problem involving the quasilinear term (|u| 2 ) u.The impulsive effects which brought from the quasilinear term (|u| 2 ) u are more complicated than u .
Motivated by the works mentioned above, in this paper, our purpose is to investigate the multiplicity of solutions for Dirichlet boundary conditions of second-order quasilinear equations with impulsive effects (1.1).Moreover, the nonlinearity f does not need to satisfy the Ambrosetti-Rabinowitz condition (see [3]).Furthermore, the impulsive terms I j (u) need to satisfy the suplinear condition rather than the sublinear condition as those in [18,23,28,29].By making use of the variant fountain theory (see [30]), the multiplicity of solutions for the problem (1.1) are obtained.

Preliminaries
In this section, the following theorem will be needed in the proof of our main results.Let E be a Banach space with the norm and Then there exist Particularly, if {u(λ n )} has a convergent subsequence for every k, then Φ 1 has infinitely many nontrivial critical points {u k } ∈ E \ {0} satisfying Φ 1 (u k ) → 0 − as k → +∞.
In the Sobolev space H 1 0 (0, T), consider the inner product inducing the norm . By Poincaré's inequality where λ = π 2 T 2 is the first eigenvalue of the problem −u = λu with Dirichlet boundary conditions, the norm u H 1 0 (0,T) and u L 2 are equivalent.But, in this paper, we define the following inner product in .
Throughout our paper, we assume that ess inf t∈[0,T] a(t) > −λ, which together with Lemma 2.1 in [29] yields that the norm u H 1 0 and u are equivalent.Thus, by the Sobolev Embedding Theorem, there exists a constant c > 0 such that u ∞ := max t∈[0,T] |u(t)| ≤ c u .
For each u ∈ H 1 0 (0, T), u is absolutely continuous and u ∈ L 2 (0, T).In this case, ∆u(t) = u (t + j ) − u (t − j ) = 0 may not hold for any t ∈ (0, T).It leads to the impulsive effects.Thus, Similarly, we have Define the functional Φ : H 1 0 (0, T) → R by where where Clearly, the critical points of Φ 1 (u) = Φ(u) correspond to the weak solutions of the problem (1.1).In H 1 0 (0, T), we can choose a completely orthonormal basis e j and set X j = Re j .Thus, Z k and Y k can be defined.

Main result
Theorem 3.1.Assume that F(t, u) is even about u and the following conditions are satisfied.
(H2) There exist constants b j > 0 and (H5) There exist constants l 2 , l 3 > 0 such that Then the problem (1.1) has infinitely many solutions.
In order to prove Theorem 3.1, we need the following lemmas.
Next, we show the proof of Theorem 3.1.