Multiplicity of Quasilinear Schrödinger Equation

In this paper, we study the quasilinear Schrödinger equation involving concave and convex nonlinearities. When the pair of parameters belongs to a certain subset of R2, we establish the existence of a nontrivial mountain pass-type solution and infinitely many negative energy solutions by using some new techniques and dual fountain theorem. Recent results from the literature are improved and extended.

In the last two decades, Equation (1) has been studied extensively due to its strong physical background. In particular, under the suitable parameters, Equation (1) can describe the self-channeling of a high-power ultra-short laser in matter, for detail [1][2][3]. In addition, Equation (1) also appears the superfluid film in plasma physics [4] and some fluid mechanics [5].
For the case where α = 1, λ = 1, and μ = 1, solutions of (1) are standing wave solutions of the following quasilinear Schrödinger equation: Many researchers focus on the nonlinear quasilinear Schrödinger Equations (1) and (2). Adachi and Watanabe [6] discussed the uniqueness of the ground state solutions for the following quasilinear Schrödinger equations: via a dual approach. The Nehari method was adopted to establish the existence results of ground state solutions by Liu et al. in [7]. The Lagrange multiplier method was used in [8]  The common point of the above works is that the nonlinear terms are all superlinear; to our best knowledge, there are few results on the quasilinear Schrödinger equation involving concave and convex nonlinearities. Furthermore, as mentioned above, there are few results on the existence of negative energy solutions. The reason is that the combination of concave and convex will make it difficult to verify the mountain pass theorem and fountain theorem. Inspired by the above-mentioned works, this paper is aimed at considering quasilinear Schrödinger Equation (1) involving concave and convex nonlinearities and proving the existence of mountain pass-type of solution and a sequence of infinitely many solutions with negative energy.
Theorem 2. Assume (V), (g 1 )-(g 3 ) and (h 1 )-(h 5 ) hold. Then, for λ > 0, μ > 0, problem (1) has a sequence of solutions with negative energy. Remark 3. (V), (g 1 ), and (h 1 )-(h 3 ) ensure the compactness of energy functional. Inspired by [12], we adopt a new approach to verify the mountain structure; on the other hand, the Tayler expansion technique plays an important role in verifying a dual fountain theorem. As mentioned above, our results may be regarded as a generalization and improvement of many existing results.
The main tools of this paper are variational methods combined with analysis techniques and suitable estimations; these methods and techniques are important for dealing with our equation and establishing the relative energy estimation. Thus, in order to make readers follow our work more easily, here, we briefly recall these helpful analysis techniques such as variational methods, iterative technique, fixed point theo-rems, and finite element methods. In [13][14][15], variational methods and some critical point theorems were employed to study the existence of solution for various elliptic equations. The iterative technique [16,17] was also developed to establish the existence criterion of solutions as well as the corresponding convergence analysis for Hessian-type elliptic equations. In addition, fixed point theorems [18][19][20][21] and finite element methods [21,22] also provide important theoretical and numerical tools for solving various nonlinear equations.
The rest of this paper is organized as follows: in Section 2, the variational framework and some lemmas are presented. Section 3 is devoted to the Proofs of Theorem 10 and Theorem 11.

Preliminaries and Functional Setting
In this paper, we make use of the following notations: Set E ≔ fu ∈ H 1 ðℝ N Þ: Ð ℝ N VðxÞu 2 < +∞g; E is a Hilbert space with inner product ðu, vÞ E = Ð ℝ N ð∇u∇v + VðxÞuvÞ. By j·j p , we denote the usual L p norm. B R denotes the open ball centered at the origin and radius R > 0: Throughout this paper, C and C i are used in various places to denote positive constants, which are not essential to the problem.
It follows from ([23], Lemma 2.1) that the embedding E ↪L r ðℝ N Þ(2 ≤ r < 2 * ) is compact, and there is η r > 0 such that juj r ≤ η r kuk for all u ∈ E: The energy functional J : E ⟶ ℝ is given by According to [6], we can make the change of variable by v = f −1 ðuÞ, where f is defined by Journal of Function Spaces After the change of variables, we obtain the following functional: which is well defined in E under the assumptions (V), (h 1 ), (h 2 ), and (g 1 ). Moreover, the critical points of the functional I correspond to the weak solutions of the following equation: It is shown in [24] that if v ∈ E is a critical point of the function I, then u = f ðvÞ ∈ E and u is a solution of (1). The function f ðtÞ enjoys some properties given in [25]. The Mountain Pass lemma in [26] allows us to find Ceramitype sequence.
(A 4 ) Φ satisfies the ðCÞ c condition for every c ∈ ½d k 0 , 0Þ. Then Φ has a sequence of negative critical values converging to 0. Remark 6. If 1 < p < 2 * , then we have
Proof of Theorem 11. Since E is a reflexive and separable Banach space, there exist fe j g ⊂ E and fe * j g ⊂ E * such that E = spanfe j : j = 1, 2,⋯g, E * = spanfe * j : j = 1, 2,⋯g in which We will use Dual Fountain theorem Lemma 5 to prove Theorem 2. Set Q is a C 2 -functional on E. Now, by the Taylor formula and some simple computation, we get On the other hand, on Z k , for kvk small enough, Ð ℝ N jGðx, f ðvÞÞj ≤ C 6 jvj r 1 2 : Thus, by Remark 6 and simple computation, there exists C > 0 such that which implies (A 1 ) holds. By (g 2 ), for kvk which is small enough, we have Since Y k is a finite dimensional space, (A 2 ) is satisfied for every γ k > 0 which small enough, when λ > 0, μ > 0. From (29), for k > k 0 , v ∈ Z k , kvk ≤ ρ k ,