# Multiplicity of Quasilinear Schrödinger Equation

Authors: Xiaorong Luo; Anmin Mao; Xiangxiang Wang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we study the quasilinear Schrödinger equation involving concave and convex nonlinearities. When the pair of parameters belongs to a certain subset ofℝ2, 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.


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## Body

## 1. Introduction and Main Results

In this paper, we are concerned with the following quasilinear Schrödinger equations:(1)−Δu+Vxu−Δu2αu2α−2u=λgx,u+μhx,u,x∈ℝN,where N≥3, 3/4<α≤1, λ and μ are the parameters, g,h∈CℝN×ℝ,ℝ, and the potential Vx satisfies the following assumption:(V) V∈CℝN,ℝ, 0<V0≔infx∈ℝNVx, and for each M>0, measx∈ℝN:Vx≤M<+∞, where V0 is a constant and meas denotes the Lebesgue measure in ℝN.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 [1–3]. In addition, Equation (1) also appears the superfluid film in plasma physics [4] and some fluid mechanics [5].For the case whereα=1, λ=1, and μ=1, solutions of (1) are standing wave solutions of the following quasilinear Schrödinger equation:
(2)iΨt+ΔΨ−WxΨ+kΔhΨ2h′Ψ2Ψ+gx,Ψ+hx,Ψ=0,x∈ℝN.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:
(3)−Δu+λu−kΔuαuα−2u=up−1u,x∈ℝN,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] to prove the existence of the ground state solutions. Recently, Liu et al. in [9] applied a perturbation method to get the existence of positive solutions and a sequence of high energy solutions. In [10], Zhang et al. established the existence of infinite solutions for a modified nonlinear Schrödinger equation by using a symmetric mountain pass theorem as well as dual approach. Recently, Zhang et al. [11] offered a new iterative technique to get the existence and nonexistence of blow-up radial solutions for a Schrödinger equation involving a nonlinear operator.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.Throughout this paper, we assume thatg and h satisfy the following assumptions:(g1) There exist constants 1<r1<r2<⋯<rm<2 and functions hix∈L2/2−riℝN,ℝ+,i=1,2,⋯,m, such that
(4)gx,u≤∑i=1mhixuri−1.(g2) There exist σ∈1,2, c1>0 such that Gx,u≥c1uσ for all x,u∈ℝN×ℝ.(g3) gx,u is odd in u.(h1) 3/4<α≤1, 4α<p<2α·2∗, hx,u≤Cu+up−1 holds for all x,u∈ℝN×ℝ.(h2) h∈CℝN,ℝ and hx,u=ou,u⟶0 uniformly on ℝN.(h3) There exists θ>4 such that
(5)0<Hx,u≔∫0uhx,sds≤uθhx,u,forallx,u∈ℝN×ℝ.(h4) There exists R>0 such that
(6)infx∈ℝN,u>RHx,u>0.(h5) hx,u is odd in u.Theorem 1.
Assume (V), (g1), and (h1)–(h4) hold. Then, there exist two positive constants μ¯,λ¯ such that for λ,μ∈0,+∞×0,μ¯∪−λ¯,λ¯×0,+∞, (1) has at least a nontrivial mountain pass-type solution.Theorem 2.
Assume (V), (g1)–(g3) and (h1)–(h5) hold. Then, for λ>0,μ>0, problem (1) has a sequence of solutions with negative energy.Remark 3.
(V), (g1), and (h1)–(h3) 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 theorems, and finite element methods. In [13–15], 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 [18–21] 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 Section2, the variational framework and some lemmas are presented. Section 3 is devoted to the Proofs of Theorem 10 and Theorem 11.


## 2. Preliminaries and Functional Setting

In this paper, we make use of the following notations:(7)H1ℝN≔u∈L2ℝN:∇u∈L2ℝN,uH12≔∫ℝN∇u2+u2,D1,2ℝN≔u∈L2∗ℝN:∇u∈L2ℝN,uD1,22≔∫ℝN∇u2.SetE≔u∈H1ℝN:∫ℝNVxu2<+∞; E is a Hilbert space with inner product u,vE=∫ℝN∇u∇v+Vxuv. By ·p, we denote the usual Lp norm. BR denotes the open ball centered at the origin and radius R>0. Throughout this paper, C and Ci 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↪LrℝN(2≤r<2∗) is compact, and there is ηr>0 such that ur≤ηru for all u∈E. The energy functional J:E⟶ℝ is given by
(8)Ju=12∫ℝN∇u2+Vxu2+14α∫ℝN∇∣u2α2−∫ℝNλGx,u+μHx,u=12∫ℝN∇u2+Vxu2+α∫ℝNu22α−1∇u2−∫ℝNλGx,u+μHx,u.According to [6], we can make the change of variable by v=f−1u, where f is defined by
(9)f′=11+2αft22α−1,f−t=−ftont∈0,+∞.After the change of variables, we obtain the following functional:(10)Iv≔Jfv=12∫ℝN∇v2+Vxf2v−∫ℝNλGx,fv+μHx,fv,which is well defined in E under the assumptions (V), (h1), (h2), and (g1). Moreover, the critical points of the functional I correspond to the weak solutions of the following equation:
(11)−Δv=11+2αfv22α−1gx,fv+hx,fvVxfvinℝN.It is shown in [24] that if v∈E is a critical point of the function I, then u=fv∈E and u is a solution of (1). The function ft enjoys some properties given in [25]. The Mountain Pass lemma in [26] allows us to find Cerami-type sequence.Lemma 4.
Assume (V), (h1), (h2), and (g1) hold. Then, I∈C1E,ℝ and
(12)I′v,w=∫ℝN∇v∇w+Vxfvf′vw−∫ℝNλgx,fvf′vw+μhx,fvf′vw,for all w∈E and ψ′:E⟶E∗ is compact, where ψv=∫ℝℕGx,fv.Proof.
The proof is standard; we omit it here.
LetX be a Banach space with the norm · and X=⊕i=1∞Xi¯ and dimXi<+∞ for each i∈ℕ. Further, we set
(13)Yk=⊕i=1kXi,Zk=⊕i=k∞Xi¯.Lemma 5 ([27]).
LetΦ∈C1X,ℝ satisfies Φ−u=Φu. Assume that, for every k>k0, there exists ρk>γk>0 such that
(A1) ak≔infu∈Zk,u=ρkΦu≥0.
(A2) bk≔maxu∈Yk,u=γkΦu<0.
(A3) dk≔infu∈Zk,u≤ρkΦu⟶0ask⟶+∞.
(A4) Φ satisfies the Cc condition for every c∈dk0,0.
ThenΦ has a sequence of negative critical values converging to 0.Remark 6.
If1<p<2∗, then we have
(14)βk≔supu∈Zk,u=1up⟶0,k⟶∞.


## 3. Proof of Main Results

Lemma 7.
Ifvn2⟶+∞, then vn02≔∫ℝN∇v2+Vxf2v⟶+∞, as n⟶∞.Proof.
If not, there exists a positive constantC>0 such that vn2⟶+∞ and vn02≤C, as n⟶∞. Then, vn02/vn2⟶0. Set wn=vn/vn and hn=f2vn/vn2, one has ∫ℝℕ∇wn2+Vxhnx⟶0. Hence,
(15)∫ℝℕ∇wn2⟶0,∫ℝℕVxhnx⟶0,∫ℝℕVxwn2x⟶1.
It follows from [25]; for each ε>0, there exists C2 independent of n such that measΩn<ε, where Ωn≔x∈ℝℕ:vnx≥C2. One has
(16)∫ℝℕ\ΩnVxwn2≤C3∫ℝℕ\ΩnVxf2vnvn2≤C3∫ℝℕVxhnx⟶0.
On the other hand, there existsε>0 such that whenever Ω′⊂RN and measΩ′<ε, ∫Ω′Vxwn2≤1/2. Thus,
(17)∫ℝℕVxwn2=∫ℝℕ\ΩnVxwn2+∫ΩnVxwn2≤o1+12,which contradicts with (15).Lemma 8.
Suppose (V), (g1), and (h1)–(h4) hold. Then, there exists two positive constants μ¯,λ¯, such that (i) for λ,μ∈0,+∞×0,μ¯∪−λ¯,λ¯×0,+∞; there exist ρ>0, η>0 such that infIv:v∈E,v=ρ>η; (ii) there exists e∈E with e>ρ such that Ie<0.Proof.
(i)
Forr>0, v∈Sr=v∈E:v=r, when r is large enough, by simple computation, one has
(18)Iv≥v0214−λC4vrm−2−μC5pvp−2.
Letms=λC4srm−2+μC5/psp−2, ms is bounded below; thus, ms admits a minimizer s0=λC4p2−rm/μC5p−21/p−rm, then
(19)infs∈0,+∞ms=ms0=μ2−rm/p−rm·λp−2/p−rmC4C4p2−rmC5p−2rm−2/p−rm+C5pC4p2−rmC5p−2p−2/p−rm.
LetK≔C4C4p2−rm/C5p−2rm−2/p−rm+C5/pC4p2−rm/C5p−2p−2/p−rm. For all μ>0 and
(20)λ≤14μ2−rm/p−rmKp−rm/p−2≔λ¯or for all λ>0 and
(21)μ≤14λp−2/p−rmKp−rm/2−rm≔μ¯,one has ms0<1/4. For v=s0, there exists s′>0 such that v0∈s′,s0. Thus, there exists ρ>0 such that infv∈SρIv>0=I0
(ii)
The conclusion is easy to check, and we omit it here.Lemma 9.
Suppose (V), (g1), and (h1)–(h3) hold. Then I satisfies the Cc conditionProof.
(1)
Letvn be a Cc sequence of I, that is, Ivn⟶c,1+vnI′vn⟶0. By Lemma 7, after some direct computation, one gets that the sequence vn is bounded
(2)
Passing to a subsequence if necessary, assume thatvn⇀v in E. First, through a proof by contradiction with the help of α∈3/4,1, there exists C4>0 such that
(22)∫ℝN∇vn−v2+Vxfvnf′vn−fvf′vvn−v≥C4vn−v2.
On the other hand, by Lemma4, ψ′ is compact, ψ′vn−ψ′v⟶0, and
(23)ψ′vn−ψ′v,vn−v⟶0asn⟶∞,
(24)limn→∞∫ℝNhx,fvnf′vn−hx,fvf′vvn−v=0.
Therefore, by (22)–, one has
(25)o1=I′vn−I′v,vn−v≥C6vn−v2+o1,which implies vn−v⟶0 as n⟶∞Proof of Theorem 10.
As a consequence of Lemmas8 and 9 and Mountain Pass theorem, we get the desired result.Proof of Theorem 11.
SinceE is a reflexive and separable Banach space, there exist ej⊂E and ej∗⊂E∗ such that E=spanej:j=1,2,⋯¯,E∗=spanej∗:j=1,2,⋯¯ in which
(26)ei,ej∗=1,i=j,0,i≠j.
SetEj=spanej,Yk=⊕i=1kEi,Zk=⊕i=k∞Ei¯. We will use Dual Fountain theorem Lemma 5 to prove Theorem 2. Set
(27)Qv≔12∫ℝN∇v2+Vxf2v,
Q is a C2-functional on E. Now, by the Taylor formula and some simple computation, we get
(28)Qv=Q0+Q′0,v+12Q″0v,v+ov2=12∫ℝN∇v2+Vxv2dx+ov2,asv⟶0.
On the other hand, onZk, for v small enough, ∫ℝNGx,fv≤C6v2r1. Thus, by Remark 6 and simple computation, there exists C>0 such that
(29)Iv≥v2C8−λC6βkr1vr1−2,which implies (A1) holds. By (g2), for v which is small enough, we have
(30)Iv=12∫ℝN∇v2+12∫ℝNVxf2v−λ∫ℝNGx,fv−μ∫ℝNHx,fv≤12v2−Cvσσ+μC2v22.
SinceYk is a finite dimensional space, (A2) is satisfied for every γk>0 which small enough, when λ>0, μ>0. From (29), for k>k0, v∈Zk, v≤ρk,
(31)Iv≥−λC6βkr1vr1≥−λC6βkr1ρkr1,since βk⟶0 and ρk⟶0 as k⟶∞; condition (A3) is also checked. (A4) follows from Lemma 9. Thus, by using Lemma 5, we get the desired results and complete the proof.



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*Source: 1894861-2020-05-11.xml*