Existence of solitary waves in nonlocal nematic liquid crystals

Nonlinear Analysis: Real World Applications 22 (2015) 107–114

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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

Existence of solitary waves in nonlocal nematic liquid crystals Guoqing Zhang a,∗ , Zhonghai Ding b a

College of Sciences, University of Shanghai for Science and Technology, Shanghai 200093, PR China

b

Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, USA

article

info

Article history: Received 3 January 2014 Received in revised form 14 August 2014 Accepted 22 August 2014

abstract In this paper, we study the solitary waves in nonlocal nematic liquid crystals. By applying the Mountain Pass Theorem and the Krasnoselskii genus theory, we prove some existence and multiplicity results of radially symmetric solitary waves. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Solitary waves Nonlocal nematic liquid crystals Variational methods

1. Introduction Liquid crystals are organic mesophases featuring various degrees of spatial order while retaining basic properties of fluid. Nematic liquid crystals (NLCs) are common materials found in many consumer electronic devices. NLCs display the longrange order of crystals, and contain rod-like molecules exhibiting both orientational alignment without position order and strong optical nonlinearities due to large refractive index anisotropy. Another important property of NLCs is the ability to change optical characteristics when an external electric field is applied and enables the macroscopic reorientation of the director tilt angle. Since the light incident on a NLC modifies the electric permittivity tensor and leads to reorientation nonlinearity, the study of propagation of self-focused beams in NLCs have attracted a lot of attentions from engineers and physicists in recent years. The optical spatial solitons in NLCs, termed as nematicons, have been observed experimentally [1–4] and numerically [5–9]. For the recent developments of nematicons, the reader is referred to an excellent review paper by Peccianti and Assanto [10]. The distortion of molecular orientation in NLCs can be described by the reorientation angle θ of the director with respect to the z-axis. In the presence of an external low frequency electric field, the spatial evolution of a slowly-varying beam envelope E (x, y, z ), which is linearly polarized along the x-axis and propagates along the z-axis, is governed by the nonlinear Schrödinger-like paraxial wave equation. The molecular orientation angle θ is governed by the elliptic equation. The model equations for the optical field (E , θ ) are given by [1,3,8] 2i

∂E + ∆x,y E + α[sin2 θ − sin2 θ0 ]E = 0, ∂z

2∆x,y θ + [β + α|E |2 ] sin(2θ ) = 0,

(1.1) (1.2)

where θ0 is the pretilt angle which is the orientation induced only by the static electric field, ∆x,y is the Laplacian, α and β denote the optical and static permittivity anisotropies of NLC molecules (see Fig. 1).

∗

Corresponding author. Tel.: +86 2165808341. E-mail addresses: [email protected], [email protected] (G. Zhang), [email protected] (Z. Ding).

http://dx.doi.org/10.1016/j.nonrwa.2014.08.006 1468-1218/© 2014 Elsevier Ltd. All rights reserved.

108

G. Zhang, Z. Ding / Nonlinear Analysis: Real World Applications 22 (2015) 107–114

Fig. 1. Sketch of the NLC cell.

The model equations (1.1) and (1.2) provide a very good agreement with experimental data [6,9], and have been studied by several researchers [5,7,11,9]. By using suitable trial function, Minzoni et al. [7] obtained the approximate modulation solutions. Strinic et al. [9] displayed numerically the existence of stable solitons in a narrow threshold region by using a fast Fourier transform algorithm. By using the modified Petviashvili method and the variational method, Aleksic et al. [5] computed the fundamental soliton profiles. Panayotaros and Marchant [11] studied the existence soliton solutions by using a concentration compactness argument, and investigated also their stability properties. Let θ = θ0 + θˆ , where θˆ corresponds to the optically induced molecular reorientation. By using the following first-order approximations:

ˆ 2 ), sin2 θ = sin2 θ0 + sin(2θ0 )θˆ + o(|θ|

(1.3)

sin(2θ ) = sin(2θ0 ) + 2 cos(2θ0 )θˆ + o(|θˆ | ), 2

(1.4)

and assuming θˆ ≪ 1, one can derive from (1.1) and (1.2) the following lower-order approximation model for the optical field (E , θˆ ) [5],

∂E + ∆x,y E + α¯ θˆ E = 0, ∂z 2∆x,y θˆ + β¯ θˆ + α| ¯ E |2 = 0,

(1.5)

2i

(1.6)

where α¯ = α sin(2θ0 ), β¯ = 2β cos(2θ0 ). By applying the nondimensionalization procedure to (1.5) and (1.6), the following dimensionless dynamical evolution system was derived and studied in [2],

∂E + ∆x,y E + γ ψ E = 0, ∂z

(1.7)

∆x,y ψ − c 2 ψ + 4π |E |2 = 0,

(1.8)

2i

where γ ≥ 0 is a parameter, c ≥ 0 is a constant. Assume a solitary wave of (1.7) and (1.8) is in the form of E (x, y, z ) = exp(iωz )u(x, y), where ω > 0 and u(x, y) is a real valued function. Then one obtains from (1.7) and (1.8) the following nonlinear elliptic system,

∆u − 2ωu + γ uψ = 0,

−∆ψ + c ψ = 4π u , 2

2

(x, y) ∈ R2 , (x, y) ∈ R . 2

(1.9) (1.10)

The study of existences of solitary wave solutions of (1.7) and (1.8) becomes the study of existences of solutions of the elliptic systems (1.9) and (1.10). One can check easily that if (u, ψ) is a solution of system (1.9) and (1.10), then u = 0 if and only if ψ = 0. Thus (u, ψ) = (0, 0) is the only trivial solution of system (1.9) and (1.10). In this paper, we study the existence of nontrivial radially symmetric solutions of the nonlinear elliptic system (1.9) and (1.10). For p > 1, let Lp (R2 ) be equipped with the norm



p

∥u∥p =

 1p

|u| dxdy R2

.

Let H 1 (R2 ) denote the usual Sobolev space with the scalar product

(u, v) =

 R2

(∇ u · ∇v + uv) dxdy,

G. Zhang, Z. Ding / Nonlinear Analysis: Real World Applications 22 (2015) 107–114

109

and the norm

 ∥ u∥ = R2

 21 (|∇ u|2 + |u|2 )dxdy .

(R ) denote the space of radially symmetric functions in H 1 (R2 ). The radial Sobolev space Hr1 (R2 ) is a closed subspace of H (R2 ). The main results of this paper are stated as follows. Let Hr1 1

2

Theorem 1.1. Let ω > 0 and γ ≤ 0. The nonlinear elliptic system (1.9) and (1.10) admits only the trivial solution u = 0 and ψ = 0 in H 1 (R2 ) × H 1 (R2 ).  Theorem 1.2. Let ω > 0 and γ > 0. The nonlinear elliptic system (1.9) and (1.10) admits at least one nontrivial solution (u, ψ) ∈ Hr1 (R2 ) × Hr1 (R2 ).  Theorem 1.2 is proved by using the Mountain Pass Theorem due to Ambrosetti–Rabinowitz [12]. In real applications or experiments, γ is a fixed physical parameter. One would be interested in the multiplicity results for the fixed γ . However, we prove in this paper a different type of multiplicity result by applying the Krasnoselskii genus theory [13]. Theorem 1.3. Let ω > 0. The nonlinear elliptic system (1.9) and (1.10) admits infinitely many triples (γn , un , ψn ) ∈ R+ × Hr1 (R2 ) × Hr1 (R2 ).  Theorem 1.3 may shed some light on the relationship between soliton solutions and parameter γ . Due to the intrinsic difference between the Mountain Pass Theorem and the Krasnoselskii genus theory, the solutions claimed in Theorem 1.3 may be, in general, different from the mountain pass type solution claimed in Theorem 1.2. This paper is organized as follows. In Section 2, we formulate a variational problem of the nonlinear elliptic system (1.9) and (1.10) and prove several lemmas. In Section 3, we prove the main theorems. 2. Variational formulation Define the functional F : H 1 (R2 ) × H 1 (R2 ) → R by F (u, ψ) = F1 (u) − γ F2 (u, ψ),

(2.1)

where F1 (u) =

1 2



(|∇ u|2 + 2ωu2 )dxdy,

R2

F2 (u, ψ) = −



1 16π

R2

(2.2)

(|∇ψ|2 + c 2 ψ 2 )dxdy +

1



2

R2

u2 ψ dxdy.

(2.3)

It is straightforward to verify that F is continuously Fréchet differentiable and the weak solutions of system (1.9) and (1.10) in H 1 (R2 ) × H 1 (R2 ) correspond to the critical points of F . Note that F (u, ψ) is strongly indefinite. It is well-known [14] that H 1 (R2 ) ⊂ Lp (R2 ) for 2 ≤ p < ∞.

(2.4)

System (1.9) and (1.10) can be reduced to a single nonlinear elliptic equation with a nonlocal term. In order to do this, we need the following lemma. Lemma 2.1. For each u ∈ H 1 (R2 ), there exists a unique ψ = ψ[u] ∈ H 1 (R2 ) which solves Eq. (1.10) and satisfies

∥ψ[u]∥ ≤ C ∥u∥2p , where 2 < p ≤ 4 and C is a constant. Moreover, ψ : H 1 (R2 ) → H 1 (R2 ) is continuous, and satisfies ψ[u] ≥ 0 on R2 and ψ[su] = s2 ψ[u] for any s ∈ R. Proof. Let 2 < p ≤ 4 and q = p−2 . Then q ≥ 2 and p

   

R2

 

u2 v dxdy ≤

 2p  |u|p dxdy

 R2

R2

2 p

+

1 q

= 1. By (2.4) and the Hölder inequality, we have

 1q |v|q dxdy ≤ C ∥u∥2p ∥v∥,

∀u, v ∈ H 1 (R2 ),

(2.5)

where C is a constant. For each u ∈ H 1 (R), upon applying the Lax–Milgram Theorem, Eq. (1.10) admits a unique weak solution ψ[u] ∈ H 1 (R2 ) such that def

a(ψ[u], v) =



 R2

 ∇ψ[u] · ∇v + c 2 ψ[u]v dxdy = 4π

 R2

u2 v dxdy,

∀v ∈ H 1 (R2 ),

110

G. Zhang, Z. Ding / Nonlinear Analysis: Real World Applications 22 (2015) 107–114

and

∥ψ[u]∥ ≤ C ∥u∥2p , where C is a constant. By (2.4), it is straightforward to verify that ψ : H 1 (R2 ) → H 1 (R2 ) is continuous. It follows from [15] that solution ψ[u] of (1.10) can be expressed by

ψ[u](x, y) =

 R2

4π E2 (c (x − s), c (y − t ))u2 (s, t )dsdt ,

(2.6)

where 1

E2 (x, y) =



2π

K0 ( x2 + y2 ),

and K0 (r ) ≥ 0 is the zero-th order modified Bessel function of second kind. Hence it is straightforward to verify that ψ[u] ≥ 0 on R2 and ψ[su] = s2 ψ[u] for any s ∈ R.  By Lemma 2.1, the nonlinear elliptic system (1.9) and (1.10) can be reduced to a single nonlinear elliptic equation,

− ∆u + 2ωu = γ uψ[u]. Define the functional Φ : H 1 (R2 ) → R by Φ (u) = F (u, ψ[u]) = F1 (u) − γ F2 (u, ψ[u]),

(2.7) (2.8)

where F , F1 and F2 are defined by (2.1)–(2.3). By Lemma 2.1, Φ is well defined and Φ ∈ C differentiation, we have

Φ (u)v = ′

 R2

(∇ u∇v + 2ωuv)dxdy − γ

 R2

uψ[u]v dxdy,

1

H (R ), R . By using the implicit



1

2



∀u, v ∈ H 1 (R2 ).

(2.9)

Hence, u ∈ H 1 (R2 ) is a weak solution of Eq. (2.7) if and only if u is a critical point of Φ . Since the embedding of H 1 (R2 ) into Lp (R2 ) is not compact, the lack of compactness makes the study of critical points of Φ in H 1 (R2 ) extremely challenging. It is well-known [16,17] that, for 2 ≤ p < ∞, the embedding of Hr1 (R2 ) into Lp (R2 ) is compact. Thus we will focus on the study of radially symmetric solutions of system (1.9) and (1.10). Let {T (k)}k∈S 1 denote the representation of the circle group S 1 on H 1 (R2 ). Then the fixed point set of this group representation, Fix(S 1 ), is given by Fix(S 1 ) = {u ∈ H 1 (R2 )|T (k)u = u, ∀k ∈ S 1 } = Hr1 (R2 ). Note that a(v, v) is invariant under {T (k)}k∈S 1 in H 1 (R2 ), that is, a(T (k)v, T (k)v) = a(v, v),

∀v ∈ H 1 (R2 ).

Thus ψ[u] ∈ Hr1 (R2 ) if u ∈ Hr1 (R2 ). It is easy to verify that functional Φ : H 1 (R2 ) → R is also invariant under {T (k)}k∈S 1 , that is,

Φ (T (k)u) = Φ (u),

∀u ∈ H 1 (R2 ), k ∈ S 1 .

By the principle of symmetric criticality of Palais [18], if u ∈ Hr1 (R2 ) is a critical point of Φ |H 1 (R2 ) , then u is a critical point of r

Φ on H 1 (R2 ). Then, a critical point of Φ |Hr1 (R2 ) is a weak radially symmetric solution of Eq. (2.7). Consequently (u, ψ), where ψ = ψ[u], is a weak radially symmetric solution of the nonlinear elliptic system (1.9) and (1.10). Lemma 2.2. Let ψ[u] be defined in Lemma 2.1. If un → u weakly in Hr1 (R2 ), then ψ[un ] → ψ[u] in Hr1 (R2 ), and

 lim

n→∞

R2

ψ( )

un u2n dxdy

 = R2

ψ(u)u2 dxdy.

(2.10)

Proof. Suppose un → u weakly in Hr1 (R2 ). Then {un } is bounded in H 1 (R2 ). Note that

      (u2 − u2 )v dxdy ≤   2 n R

4

1/4 

4

|un + u| dxdy R2

1/4

|un − u| dxdy R2

∥v∥2 ,

∀v ∈ H 1 (R2 ).

By (2.4), we have

      (u2 − u2 )v dxdy ≤ C  2 n  R

1/4 |un − u|4 dxdy ∥v∥, R2

∀v ∈ H 1 (R2 ),

where C is a constant. Note that a(ψ[un ] − ψ[u], v) =

 R2

(u2n − u2 )v dxdy,

∀v ∈ H 1 (R2 ).

(2.11)

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111

By applying the Lax–Milgram Theorem, we have

∥ψ[un ] − ψ[u]∥ ≤ C ∥un − u∥4 . Since Hr1 (R2 ) ↩→ Lp (R2 ) is compact for 2 ≤ p < ∞, we have un → u in L4 (R2 ). We then have ψ[un ] → ψ[u] in Hr1 (R2 ). Consequently, (2.10) follows from (2.11).  Define the functional G : Hr1 (R2 ) → R by G(u) = F2 (u, ψ[u]) = −

1 16π

 R2

(|∇ψ[u]|2 + c 2 |ψ[u]|2 )dxdy +

1 2

 R2

u2 ψ[u]dxdy.

(2.12)

Lemma 2.3. (a) G is continuously Fréchet differentiable in Hr1 (R2 ), that is, G ∈ C 1 Hr1 (R2 ), R . (b) For σ > 0, define set Mσ by





Mσ = {u ∈ Hr1 (R2 )|G(u) = σ }. Then Mσ is a non-empty C 1 manifold of codimension 1. Proof. By the implicit differentiation, we have

∂ F2 (u, ψ[u]) = 0, ∂ψ

∀u ∈ Hr1 (R2 ).

(2.13)

It is easy to verify that G(u) is continuously Fréchet differentiable in Hr1 (R2 ), and

∂ F2 (u, ψ[u]) ′ ∂ F2 (u, ψ[u]) ∂ F2 (u, ψ[u]) v+ ψ [u]v = v ∂u ∂ψ ∂u  = uψ[u]v dxdy, ∀u, v ∈ Hr1 (R2 ).

G′ (u)v =

(2.14)

R2

Since

 R2

(|∇ψ[u]|2 + c 2 |ψ[u]|2 )dxdy = 4π

 R2

u2 ψ[u]dxdy,

we have G(u) =

1 4

 R2

u2 ψ[u]dxdy.

(2.15)

Thus Mσ can be expressed as

 Mσ = u ∈

Hr1

  1 (R )  4 2

R2

u2 ψ[u]dxdy = σ



.

To prove Mσ is nonempty, define K : R+ → R, for any given u ∈ Hr1 (R2 ) and u ̸= 0, by K (s) = G(su) =

s4 4

 R2

u2 ψ[u]dxdy.

Then K (0) = 0, and K (s) is monotone increasing on R+ . Thus, there exists s0 > 0 such that K (s0 ) = G(s0 u) = σ . Hence Mσ is nonempty. By Lemma 2.1, (2.14) and (2.15), we obtain that if u ∈ Hr1 (R2 ) satisfies G′ (u) = 0, then G(u) = 0, which implies u = 0. Hence, Mσ is a C 1 manifold of codimension 1.  Next, we recall the Krasnoselskii genus theory [13]. Let E be a real Banach space. Let A denote the set of all closed subsets A ⊂ E \ {0} that are symmetric with respect to the origin, that is, x ∈ A implies −x ∈ A. For A ∈ A, the Krasnoselskii genus γ0 (A) of A is defined by the least positive integer k such that there is an odd mapping h ∈ C (A, Rk \ {0}). If such a k does not exist, we set γ0 (A) = +∞. If A = ∅, then γ0 (A) = 0. If Ω ⊂ RN is an open bounded and symmetric set and 0 ∈ Ω , then γ0 (∂ Ω ) = N. Lemma 2.4. [13] Let E be a real Banach space. Suppose that I is an even C 1 -function on a complete symmetric C 1,1 manifold M ⊂ E \ {0}. Suppose that I satisfies (PS) condition and is bounded from below on M . Let

γ˜0 (M) = sup{γ0 (K ) : K ⊂ M is compact and symmetric} ≤ ∞. Then I admits at least γ˜0 (M ) pairs of critical points on M .



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G. Zhang, Z. Ding / Nonlinear Analysis: Real World Applications 22 (2015) 107–114

3. Proof of main theorems Proof of Theorem 1.1. Assume γ ≤ 0. Let (u, ψ) ∈ H 1 (R2 ) × H 1 (R2 ) be a weak solution for the system (1.9) and (1.10). Multiply Eq. (1.9) by u and integrate both sides on R2 ,



(|∇ u| + 2ωu )dxdy − γ 2

0 = R2

 ≥ R2

2

 R2

u2 ψ dxdy

(|∇ u|2 + 2ωu2 )dxdy,

where we have used the fact ψ ≥ 0 by Lemma 2.1. Since ω > 0, we have u = 0, hence ψ = 0 by Eq. (1.10).



We prove Theorem 1.2 by using the Mountain Pass Theorem [12]. Proof of Theorem 1.2. Note that Φ (0) = 0. By Lemma 2.1, there exists a constant C > 0 such that

Φ (u) =

≥

1 2 1 2



(|∇ u| + 2ωu )dxdy − 2

R2

2

γ



4

R2

u2 ψ[u]dxdy

min{1, 2ω}∥u∥2 − C γ ∥u∥4 .

Then there exists constants ρ > 0 and r0 > 0 such that 0 is a minimum of Φ (u) in Bρ (0) and

Φ (u) ≥ r0 for u ∈ ∂ Bρ (0). Let u0 ∈ Hr1 (R2 ) and u0 ̸= 0. For s > 0, we have

Φ (su0 ) =

s2 2

 R2

(|∇ u0 |2 + 2ωu20 )dxdy −

γ s4 4

 R2

u20 ψ[u0 ]dxdy.

Since ω > 0 and γ > 0, we have ψ(su0 ) → −∞ as s → ∞. Thus there exist s0 > 0 such that s0 u0 ̸∈ Bρ (0) and Φ (s0 u0 ) < 0 < r0 . Next we prove functional Φ : Hr1 (R2 ) → R satisfies the Palais–Smale condition. Let d ∈ R and {un } be a sequence in 1 Hr (R2 ) such that

Φ (un ) → d and Φ ′ (un )un → 0. Then

 lim

n→∞

R2

   (|∇ un |2 + 2ωu2n )dxdy = lim 4Φ (un ) − Φ ′ (un )un = 4d. n→∞

Thus {un } is bounded in Hr1 (R2 ). By the weak compactness of Hr1 (R2 ), {un } has a subsequence, which is denoted again by itself, such that un → u weakly in Hr1 (R2 ). Since the embedding of Hr1 (R2 ) in Lp (R2 ) is compact for any p ≥ 2, we have un → u a.e. in R2 . By Lemma 2.2 and Φ ′ (un )(un − u) → 0, that is,

 R2

∇ un ∇(un − u)dxdy + 2ω

 R2

un (un − u)dxdy − γ

 R2

ψ[un ]un (un − u)dxdy → 0,

we have un → u in Hr1 (R2 ). Thus, by applying the Mountain Pass Theorem [12], Ψ possesses a critical value d ≥ r0 , which can be characterized as d = inf

max Φ (u),

g ∈Γ u∈g ([0,1])

where Γ = {g ∈ C ([0, 1], H01 (R2 ))|g (0) = 0, g (1) = s0 u0 }. Hence Φ has at least one nontrivial critical point u ∈ Hr1 (R2 ), which is a weak solution of (2.7). Consequently, the nonlinear elliptic system (1.9) and (1.10) admits at least one nontrivial radially symmetric solution (u, ψ[u]) ∈ Hr1 (R2 ) × Hr1 (R2 ).  Let H = {u ∈ Hr1 (R2 )|u ≥ 0}. Then H is a closed subspace of Hr1 (R2 ). By restricting functional Φ on H, we have the following corollary. Corollary 3.1. Let ω > 0 and γ > 0. The nonlinear elliptic system (1.9) and (1.10) admits at least one positive solution (u, ψ) ∈ H × H. Proof. By repeating the same argument as the proof of Theorem 1.2, we obtain that Φ has at least one nontrivial critical point u ∈ H, which is nonnegative and a weak solution of (2.7). To prove u is positive on R2 , a simple application of the strong maximum principle [19] gives u > 0 in R2 . Hence system (1.9) and (1.10) has at least one positive radially symmetric weak solution in Hr1 (R2 ) × Hr1 (R2 ). 

G. Zhang, Z. Ding / Nonlinear Analysis: Real World Applications 22 (2015) 107–114

113

We use Lemma 2.4 to prove Theorem 1.3. Proof of Theorem 1.3. Let σ > 0 be given. Consider the functional F1 |Mσ , where F1 is defined by (2.2) and Mσ is defined in Lemma 2.3. Since ω > 0, we have F 1 ( u) =

1



2

R2

(|∇ u|2 + 2ωu2 )dxdy ≥

1 2

min{1, 2ω}∥u∥2 > 0.

Hence F1 is bounded from below on Mσ . Suppose un → u weakly in Hr1 (R2 ). By Lemma 2.2 and (2.15), we have lim G(un ) = lim

n→∞

n→∞

1 4

 R2

u2n ψ[un ]dxdy =



1 4

R2

u2 ψ[u]dxdy = G(u).

Hence, Mσ is weakly closed in Hr1 (R2 ). Next we prove F1 |Mσ satisfies the Palais–Smale condition. Let {un } ⊂ Mσ be a sequence such that (i) {F1 (un )} is bounded; and (ii) F1′ |Mσ (un ) → 0 as n → ∞, where F1′ |Mσ denote the derivative of the restriction of F1 to Mσ . Then, by the Lagrange multiplier method [20] and the Bolzano–Weierstrass Theorem [21], there exist a subsequence of {un }, denoted by {un } again, a sequence {γn } ⊂ R and d ∈ R, such that

 R2

(∇ un ∇v + 2ωun v)dxdy − γn

 R2

un ψ[un ]v dxdy → 0,

∀v ∈ Hr1 (R2 ),

(3.1)

and F 1 ( un ) =

 R2

(|∇ un |2 + 2ωu2n )dxdy → d,

as n → ∞.

(3.2)

Since ω > 0, it follows from (3.2) that {un } is bounded in Hr1 (R2 ). By letting v = un in (3.1), we have

 R2

(|∇ un |2 + 2ω|un |2 )dxdy − γn

 R2

u2n ψ[un ]dxdy → 0,

as n → ∞.

(3.3)

By (2.4) and Lemma 2.1, ψ[un ] is bounded in Hr1 (R2 ). Note that

 R2

u2n ψ[un ]dxdy = 4σ > 0,

∀n ≥ 1.

It follows from Eqs. (3.2) and (3.3) that {γn } is convergent and

γ = lim γn = n→∞

d 4σ

> 0.

Since {un } is bounded in Hr1 (R2 ), we may assume that un → u weakly in Hr1 (R2 ). By Lemma 2.2, ψ[un ] → ψ[u] in Hr1 (R2 ). If u = 0, then we have 4σ = lim

n→∞

 R2

un ψ[un ]dxdy = 0,

which contradicts σ > 0. Thus u ̸= 0. By letting v = un − u in (3.1), we have

 R2

∇ un ∇(un − u)dxdy +

 R2

2ωun (un − u)dxdy − γn

 R2

un ψ[un ](un − u)dxdy → 0.

(3.4)

Since the embedding of Hr1 (R2 ) into Lp (R2 ) is compact for p ≥ 2, we have un → u in Lp (R2 ) for p ≥ 2. Then, by Lemma 2.2, the second and third terms of (3.4) converge to 0 as n → ∞. Thus, it follows from (3.4) that un → u in Hr1 (R2 ). Next, we prove γ˜0 (Mσ ) = ∞. Consider a sequence {Hm } ⊂ Hr1 (R2 ), where Hm is a subspace of Hr1 (R2 ) with dimension m. Let Km = Hm ∩ Mσ . We claim Km is bounded. If not, there exists a unbounded sequence {un } ⊂ Km such that ∥un ∥ → ∞. Consider now the sequence {wn } where wn = ∥uun ∥ . Since ∥wn ∥ = 1 for n ≥ 1, it has a weakly convergent subsequence in n

Hr1 (R2 ), which is denoted by itself again, such that

wn → w weakly in Hr1 (R2 ). By Lemma 2.2, we have

 lim

n→∞

R2

wn2 ψ [wn ] dxdy =

 R2

w 2 ψ[w]dxdy.

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G. Zhang, Z. Ding / Nonlinear Analysis: Real World Applications 22 (2015) 107–114

Note that

 R2

u2n ψ[un ]dxdy = 4σ .

By dividing both sides of this equation by ∥un ∥4 and letting n → ∞, we have

 R2

w 2 ψ[w]dxdy = 0.

Thus w = 0 on R2 , which is a contradiction since ∥wn ∥ = 1 for n ≥ 1 and Km is finite dimensional. Hence we proved Km is bounded. Let S m−1 be a unit sphere in Hm . Define the mapping χ : S m−1 → Km by

χ (u) = λ∗ (u)u, where λ∗ (u)u is the unique point of the line {λu|λ ≥ 0} intersecting Mσ . Since Mσ is a non-empty manifold of codimension 1, χ is well-defined on S m−1 . It is straightforward to verify that χ is an odd continuous map. By Proposition 5.2 in [13], we have γ˜0 (S m−1 ) = m. Hence, we have γ˜0 (Km ) ≥ m. Since Km = Hm ∩ Mσ ⊂ Mσ , we obtain γ0 (Mσ ) ≥ γ˜0 (Km ) ≥ m. Since m is arbitrary, we have γ0 (Mσ ) = ∞. Therefore, by Lemma 2.4, F1 |Mσ admits infinitely many critical points on Mσ . Consequently, the nonlinear elliptic system (1.9) and (1.10) admits infinitely many triples (γn , un , ψn ) ∈ R+ × Hr1 (R2 ) × Hr1 (R2 ).  Acknowledgments We would like to thank the referees for detailed comments and suggestions to improve the quality and clarity of this paper. This research is supported by National Project Cultivate Foundation of USST (No. 13XGM05), Hujiang Foundation of China (B14005), National Natural Science Foundation of China (No. 11471215) and the Shanghai Leading Academic Discipline Project (No. XTKX2012). G. Zhang wishes to express his gratitude to the Department of Mathematical Sciences of the University of Nevada Las Vegas (UNLV) for its hospitality. This paper was completed during his visit to the UNLV from January 2013 to January 2014. References [1] C. Conti, M. Peccianti, G. Assanto, Route to nonlocality and observation of accessible solitons, Phys. Rev. Lett. 91 (2003) 073901. [2] W. Hu, T. Zhang, Q. Guo, L. Xuan, S. Lan, Nonlocality controlled interaction of spatial solitons in nematic liquid crystals, Appl. Phys. Lett. 89 (2006) 071111. [3] M. Peccianti, C. Conti, G. Assanto, A. De Luca, C. Umeton, Routing of anisotropic spatial solitons and modulational instability in liquid crystals, Nature 432 (2004) 733–737. [4] C. Rotschild, O. Cohen, O. Manela, M. Segev, T. Carmon, Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons, Phys. Rev. Lett. 95 (2005) 213904. [5] N. Aleksic, M. Petrovic, A. Strinic, M. Belic, Solitons in highly nonlocal nematic liquid crystals: Variational approach, Phys. Rev. A 85 (2012) 033826. [6] J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, M. Haelterman, Simulations and experiments on self-focusing conditions in nematic liquid-crystal planar cells, Optics Express 12 (2004) 1011–1018. [7] A. Minzoni, N. Smyth, A. Worthy, Modulation solutions for nematicon propagation in nonlocal liquid crystals, J. Opt. Soc. Amer. B Opt. Phys. 24 (2007) 1549–1556. [8] C. Reimbert, A. Minzoni, N. Smyth, Spatial soliton evolution in nematic liquid crystals in the nonlinear local regime, J. Opt. Soc. Amer. B Opt. Phys. 23 (2006) 294–301. [9] A. Strinic, D. Jovic, M. Petrovic, D. Timotijevic, N. Aleksic, M. Belic, Spatiotemporal instabilities of counterpropagating beams in nematic liquid crystals, Opt. Mater. 30 (2008) 1213–1216. [10] M. Peccianti, G. Assanto, Nematicons, Phys. Rep. 516 (2012) 147–208. [11] P. Panayotaros, T.R. Marchant, Solitary waves in nematic liquid crystals, Physica D 268 (2014) 106–117. [12] A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [13] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, fourth ed., Springer-Verlag, Berlin, 2008. [14] R. Adams, Sobolev Spaces, Academic Press, New York, 1978. [15] C. Stuart, An introduction to elliptic equations on RN , in: A. Ambrosetti, K. Chang, I. Ekeland (Eds.), Nonlinear Functional Analysis and Applications to Differential Equations, World Scientific Press, Singapore, 1998, pp. 237–285. [16] W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977) 149–162. [17] M. Willem, Minimax Theorems, Birkhauser, Boston, 1996. [18] R. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979) 19–30. [19] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1998. [20] J.-P. Aubin, Applied Functional Analysis, John Wiley & Sons, 1979. [21] H. Royden, P. Fitzpatrick, Real Analysis, fourth ed., Pearson Education, 2010.