Multiple solutions for a modified quasilinear Schrödinger elliptic equation with a nonsquare diffusion term

Xinguang Zhanga,b,1 , Lishan Liu , Yonghong Wu , Benchawan Wiwatanapataphee School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, Shandong, China zxg123242@163.com Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia y.wu@curtin.edu.au; b.wiwatanapataphee@curtin.edu.au School of Mathematical Sciences, Qufu Normal University, Qufu, 273165 Shandong, China mathlls@163.com


Introduction
Let Ω be a nonempty bounded open set of the real Euclidean space R N (N 2) with C 1boundary ∂Ω, consider the multiple solutions for the following quasilinear Schrödinger elliptic equation with the p-Laplacian and nonsquare diffusion term: where ∆ p u = div(|∇u| p−2 ∇u), N < p 2α, λ 0 is a parameter, f : Ω × R → R is a continuous function.
Equation (1) involves a quasilinear and nonconvex diffusion term ∆ p (|u| 2α ) |u| 2α−2 u. In the literature, it is referred as so-called modified nonlinear Schrödinger equation. For the case p = 2, the solution of (1) is related to standing wave solutions of the following quasilinear Schrödinger equation: where z : R × R n → C, V : R n → R is a given potential, h and g are real functions, κ is a real constant. Putting z(t, x) = e −iβt u(x) in (2), where β ∈ R and u(x) > 0 is a real function, the quasilinear equation (2) reduces to the following modified elliptic form: If h(s) = s, then (3) turns into a superfluid film equation in plasma physics Kurihara [8] used this equation to model the time evolution of the condensate wave function in superfluid film. Moreover, if h(s) = (1 + s) 1/2 , equation (3) is transformed to the following elliptic form: which is a model of the self-channeling of a high-power ultrashort laser in matter [7,16].
Many mathematical methods, such as dual approach [3,22,[24][25][26], iterative techniques [19,[27][28][29][30], fixed point theorem [5,14,21], variational methods [6,15,23] and normal boundary intersection method [9,10], have been employed to solve the properties and control problems for various differential equations. In particular, by using a constrained minimization argument Poppenberg et al. [15] established the existence of positive ground state solution for quasilinear Schrödinger equation (4). Colin and Jeanjean [3], João Marcos and Severo [6] studied the existence of positive solutions for (4) by the change of variables. The Nehari method and the symmetric mountain pass lemma were also used to establish the existence of solutions in [2,4,11]. In [13], Liu et al. studied the following quasilinear Schrödinger equation: where λ 0, 4α < p where "meas" denotes the Lebesgue measure in R N . Condition ( V) is an essential assumption, which guarantees that the embedding Clearly, assumption ( V) fails to hold for a general continuous and bounded function. Thus, if the potential V (x) fails to satisfy ( V), whether the multiple solutions of problem (5) still exist or not? In order to answer this question, in this paper, we investigate the more general modified nonlinear Schrödinger equation (1) and get a positive answer, i.e., if the potential V (x) is a general continuous and bounded function, then there exist the multiple solutions to the quasilinear Schrödinger elliptic equation with the p-Laplacian and nonsquare diffusion term (1) under suitable growth conditions. The rest of this paper is organized as follows. In Section 2, with help of a change of variables, we set up the variational framework for problem (1) and give some lemmas of the functional associated with problem (1). In Sections 3 and 4, by using Riccer's critical point theorem we give the proof of main results.

Dual approach
Let E = W 1,p (Ω) (p 1) be the Sobolev spaces with the norm We focus on the existence of nontrivial weak solutions of problem (1). A function u is called a weak solution of problem (1) if u ∈ W 1,p 0 (Ω) and for any ϕ ∈ C ∞ 0 (Ω), one has Ω 1 + (2α) p−1 |u| (2α−1)p |∇u| p−2 ∇u∇ϕ But we notice that the natural functional of problem (1) may not be well defined and not Gâteaux-differentiable in the corresponding Sobolev space E.
https://www.journals.vu.lt/nonlinear-analysis Thus, inspired by [12], we define a function h by , t 0, Let u = h(v), then Moreover, the corresponding energy functional J(v) is well defined on W 1,p (Ω). Since is a critical point of the functional J, i.e, for any ϕ ∈ W 1,p (Ω), then v is a weak solution of the equation Thus, from (6) and (7) it is easy to know that u = h(v) is a weak solution of problem (1). As the result, it is sufficient to consider the existence of solutions of (7) in W 1,p (Ω).
The following lemma can be found in [2]: The function h(t) enjoys the following properties: (h1) h ∈ C 2 is uniquely defined, odd, increasing and invertible in R; (h9) For each τ > 0, there exists where u ∞ = sup x∈Ω |u(x)|. Different from [4,11,13], the following assumption on potential is adopted in this paper: Now define two functionals Φ, Ψ : E → R as follows: For any v, w ∈ E, we have Φ, Ψ ∈ C 1 (E, R) and Proof. Let v = 0, otherwise, the conclusion holds. In the following, we argue by contradiction to prove (9). Suppose that there exists a sequence {v n } ⊂ E satisfying v n = 0 for all n ∈ N such that Set w n = v n / v n , then w n = 1. Noticing the compactness of embedding E → L s for s ∈ [1, +∞) up to a subsequence, we have w n (x) on Ω. It follows from (10) and Now according to the strategy in [26], we claim that for any ε > 0, there exists a constant τ > 0 independent of n such that meas(B n := {x ∈ Ω: |v n | τ }) ε, where meas(·) denotes the standard Lebesgue measure.
In fact, if not, there exists ε 0 > 0 such that meas(A n ) ε 0 , where A n = {x ∈ Ω: |v n | n}. By (h8) and the Fatou lemma we get The above fact contradicts with the boundedness of {Φ(v n )}. Therefore, the above conclusion is valid. Next, it follows from the Hölder inequality and the Sobolev embedding theorem that there exists ε small enough such that where C 1 is a constant, which is independent of ε.
Proof. Firstly, by (h4) and (h7) of Lemma 1 we have which implies that for any sufficiently small > 0, there exists a constant C > 0 such that On the other hand, for any v ∈ E with v > 1, (h10) of Lemma 1 and (14) yield Notice that E → L s for s ∈ [p, p * ) is continuous, then for any v ∈ E with v > 1, choose sufficiently small ε such that It follows from N < p 2α, N 2, (15) and (16) which implies that Φ is coercive. The fact that Φ is hemicontinuous can be verified using standard arguments. In addition, with the help of Theorem 26(A) in [20], as well as J(v) = I(h(v)) and the inequality we know that Φ exists and is continuous. https://www.journals.vu.lt/nonlinear-analysis

The existence of three solutions
In this section, we show the existence of three solutions of (1). The main tool used for analysis is the Riccer's critical point theorem [1,18], which is given below for reader's convenience.
Before stating our main results, we firstly denote two constants where c, V 0 , V 1 and are defined by (8), (V) and Lemma 1, |Ω| is the Lebesgue measure of Ω. Then some assumptions on F (x, s) to be used are also listed below: (F1) There exist a function a(x) ∈ L 1 (Ω) and 0 < σ < p such that for all (x, s) ∈ Ω × R, F (x, s) a(x)(1 + |s| σ ); (F2) F (x, 0) = 0 for any x ∈ Ω; (F3) There exists t 0 ∈ R with |t 0 | > 1 such that Now we state our main result here.  ∞) and a positive real number ρ > 0 such that for any λ ∈ Λ, the quasilinear elliptic equation (1) has at least three weak solutions whose norms are less than ρ.
Proof. By the definitions of Φ and Ψ we know that Ψ is compact and Φ is weakly lower semicontinuous. Further, from Lemma 3 we know that (Φ ) −1 is well defined and continuous. Now we show that the hypotheses of Lemma 4 are fulfilled.

Thus, (ii) of Lemma 4 is satisfied.
On the other hand, from (F2) and (F3) we get Ω F (x, t 0 ) dx 0 and Next, we focus our attention on the case when v ∈ E with Φ(v) r. By (7) and (8) From (17), (18) and (F3) it is easy to get that condition (iii) of Lemma 4 holds. Thus, all the hypotheses of Lemma 4 are satisfied, and hence, according to Lemma 4, there exist an open interval Λ ⊂ (0, ∞) and a positive real number ρ > 0 such that for any λ ∈ Λ, the quasilinear elliptic equation (1) has at least three weak solutions whose norms are less than ρ. and lim |z|→∞ F (x, z) > 0 for x ∈ Ω uniformly holds.
Then there exist an open interval Λ ⊂ (0, ∞) and a positive real number ρ > 0 such that for any λ ∈ Λ, the quasilinear elliptic equation (1) has at least three weak solutions whose norms are less than ρ.
Proof. By (F1), similar as the proof of Theorem 1, it is easy to know that hypothesis (i) of Lemma 4 holds. Thus, we only need to verify hypotheses (ii) and (iii). In fact, it follows from (F3 * ) that for any x ∈ Ω, there exists a sufficiently large

Thus, hypothesis (ii) of Lemma 4 is satisfied.
On the other hand, from (F2) and (F3 * ) we have Moreover, for Φ(v) r, v ∈ E, by (8) and Lemma 2 we have The above inequality and (h3) of Lemma 1 show that (18) and (19) show that condition (iii) of Lemma 4 holds. According to Lemma 4, the conclusion of Theorem 2 also holds.

The existence of infinitely many solutions
In this section, we use an infinitely many critical points theorem to obtain the multiple solutions result of problem (1). Let E be a reflexive real Banach space, Φ : E → R be a (strongly) continuous, coercive sequentially weakly lower semicontinuous and Gâteaux-differentiable functional, Ψ : E → R be a sequentially weakly upper semicontinuous and Gâteaux-differentiable functional.
Suppose f : Ω × R → R + is continuous and denote We state the result of the multiple solutions as follows: Theorem 3. Assume that l/L < p /(c p V 1 |Ω|) hold. Then for any λ ∈ (V 1 |Ω|/(pL), /(c p l), the quasilinear elliptic equation (1) has an unbounded sequence of weak solutions in W 1,p (Ω).
Proof. Firstly, for any v ∈ E, define Then Φ : E → R is a continuous, coercive sequentially weakly lower semicontinuous and Gâteaux-differentiable functional, Ψ : E → R is a sequentially weakly upper semicontinuous and Gâteaux-differentiable functional.
Now we show that the functional Φ − λΨ is unbounded from below. To do this, we take a real sequence {e n } such that lim n→∞ e n = +∞. Noticing (h8) of Lemma 1, we have h(e n ) b 0 e 1/(2α) n → ∞, n → ∞, and then Let w n (x) = e n , n ∈ N, x ∈ Ω, then we have and We divide into two cases for L to prove that Φ − λΨ is unbounded from below. Case 1. If L < +∞, choose 0 < < L − V 1 |Ω|/(λp), then by (21) there exists N 0 > 0 such that for any n > N 0 , we have Thus, It follows from the choice of that V 1 |Ω|/p − λ(L − ) < 0, and then one gets lim n→∞ (Φ(w n ) − λΨ (w n )) = −∞. The above facts show that the functional Φ−λΨ is unbounded from below. According to (i) of Lemma 5, the functional Φ − λΨ admits a sequence {v n } of critical points, that is, {h(v n )} are exactly the weak solutions of the quasilinear elliptic equation (1).