EXISTENCE AND NONEXISTENCE OF NONTRIVIAL WEAK ...

EXISTENCE AND NONEXISTENCE OF NONTRIVIAL WEAK SOLUTION FOR A CLASS OF GENERAL CAPILLARITY SYSTEMS G.A. AFROUZI, N.T. CHUNG∗ , Z. NAGHIZADEH

Abstract. The goal of this paper is to study the nonexistence and existence of nonnegative, nontrivial weak solution for a class of general capillarity systems. The proofs rely essentially on the minimum principle combined with the mountain pass theorem.

1. Introduction Capillarity can be briefly explained by considering the effects of two opposing forces: adhesion, i.e. the attractive (or repulsive) force between the molecules of the liquid and those of the container; and cohesion, i.e. the attractive force between the molecules of the liquid. The study of capillary phenomena has gained some attention recently. This increasing interest is motivated not only by fascination in naturally-occurring phenomena such as motion of drops, bubbles, and waves but also its importance in applied fields ranging from industrial and biomedical and pharmaceutical to microfluidic systems, see [12, 13]. In this paper, we are concerned with the following nonlinear elliptic system    − div(h1 (|∇u|p )|∇u|p−2 ∇u) = λFu (x, u, v) in Ω,    − div(h2 (|∇v|q )|∇v|q−2 ∇v) = λFv (x, u, v) in Ω,      u=v = 0 in ∂Ω,

(1.1)

where Ω be a bounded domain with smooth boundary ∂Ω in RN (N ≥ 3), 2 ≤ p, q < N , h1 , h2 ∈ C(R, R) and (Fu , Fv ) = ∇F stands for the gradient of F , λ is a parameter. It should be noticed that if h1 (t) = h2 (t) = 1 +

√ t , 1+t2

t ≥ 0, then (1.1) is called a capillarity system.

To our knowledge, elliptic equations of (1.1) type has been firstly investigated by J. M. ´ [14], in which the author extended the existence results by D.G. Costa et al. Bezerra do O [7] (for the p-Laplacian) to a more general class of operators. He also achieved a multiplicity Date: December 10, 2011; Revised: April 17, 2012. Key words and phrases. Weak solutions; Nonexistence, Multiplicity, Capillarity systems; Variational methods ∗

Corresponding author 2000 Mathematics Subject Classifications: 35B30, 35J60, 35P15. 1

2

G.A. AFROUZI, N.T. CHUNG, Z. NAGHIZADEH

result using Morse theory. On this topic, we refer to recent interesting papers [8, 9, 10, 11, 18]. There, the authors have used different methods to prove the existence of a nontrivial solution or the existence of infinitely many solutions. In [1, 19], the authors studied the existence of a solutions for (1.1) using the minimum principle. The goal of this paper is to give some sufficient conditions on the pontentials hi , i = 1, 2 and the nonlinearity F to get the nonexistence and the existence of at least two solutions for (1.1). Our paper extends or complements the previous results eventually in the case hi ≡ 1, i = 1, 2 (see [3, 4, 5, 15]). Throughout this paper for (t, s) ∈ R2 , we denote |(t, s)|2 = |t|2 + |s|2 . We assume that F : Ω × R × R → R is a C 1 −function, satisfying the following conditions: (H1) F (x, 0, 0) = 0 for a.e. x ∈ Ω, F (x, t, s) = F (x, 0, s) for all t ≤ 0, s ∈ R and a.e. x ∈ Ω, F (x, t, s) = F (x, t, 0) for all t ∈ R, s ≤ 0 and a.e. x ∈ Ω; (H2) It holds that q(p−1) p(q−1) |Ft (x, t, s)| ≤ C 1 + |t|p−1 + |s| p and |Fs (x, t, s)| ≤ C 1 + |t| q + |s|q−1 for all t, s ∈ R and a.e. x ∈ Ω. We say that a function γ verifies the property (Γ) if and only if γ(t, s) ≤ M (|t|p + |s|q )

(Γ)

for all t, s ∈ R, where M > 0 is independent of γ. Let Gi , i = 1, 2 be two functions satisfying property Γ. Motivated by an eigenvalue problem considered in [4], we introduce the following assumptions on the behavior of F at origin and at infinity: (H3) It holds that F (x, t, s) ≤0 |(t,s)|→0 G1 (t, s)

lim sup uniformly in x ∈ Ω; (H4) It holds that

F (x, t, s) ≤0 |(t,s)|→∞ G2 (t, s) lim sup

uniformly in x ∈ Ω. Regarding the functions h1 , h2 , we assume that (H5) hi : [0, +∞) → R, i = 1, 2 are continuous and there exist αi , βi > 0, such that αi ≤ hi (t) ≤ βi for all t ≥ 0.

A CLASS OF GENERAL CAPILLARITY SYSTEMS

3

(H6) There are constants γi > 0, i = 1, 2, such that h1 (|ξ|p )|ξ|p−2 ξ − h1 (|η|p )|η|p−2 η · (ξ − η) ≥ γ1 |ξ − η|p , h2 (|ξ|q )|ξ|q−2 ξ − h2 (|η|q )|η|q−2 η · (ξ − η) ≥ γ2 |ξ − η|q for all ξ, η ∈ RN . Remark 1.1. It should be noticed that the functions h1 (t) = h2 (t) = 1 +

√ t , 1+t2

t ≥ 0 satisfy

the condition (H6), see the Appendix. This says that (1.1) is a capillarity system. We also see that the condition (H2) was used by Zhao and Tang [20], in which the authors considered the special case h1 (t) = h2 (t) ≡ 1, G1 (t, s) = G2 (t, s) = |t|p + |s|q and studied the existence of solutions for resonance problems of (p, q)-Laplacian type. Let H01,p (Ω) and H01,q (Ω) be the usual Sobolev spaces with respect to the norms Z 1 Z 1 p q and kvk1,q = |∇v|q dx , kuk1,p = |∇u|p dx Ω

Ω

respectively and set H =

H01,p (Ω)

×

H01,q (Ω).

Then H is a Banach space under the norm

kwkH = kuk1,p + kvk1,q ,

w = (u, v) ∈ H

and the continuous embedding H ,→ Li (Ω) × Lj (Ω) holds for 1 ≤ i ≤ Moreover, this embedding is compact for all 1 ≤ i
0 such that for all λ < λ∗ , system (1.1) has no nontrivial weak solution. Theorem 1.4. Suppose that (H1)-(H6) are satisfied. Moreover, we assume that there exist B ⊂ Ω and t0 , s0 > 0, such that F (x, t0 , s0 ) > 0 for all x ∈ B. Then, there exists a constant λ∗ > 0 such that system (1.1) has at least two distinct, nonnegative, nontrivial weak solutions, provided that λ ≥ λ∗ . 2. Proofs of the main results Proof of Theorem 1.3. In what follows, we denote λp,q = min{λp , λq }, where λp and λq are the first eigenvalues of (−∆p , H01,p (Ω)) and (−∆q , H01,q (Ω)), respectively. Then we have for all u ∈ H01,p (Ω) and v ∈ H01,q (Ω), Z Z Z Z p p q λp |u| dx ≤ |∇u| dx and λq |v| dx ≤ |∇v|q dx. Ω

Ω

Ω

It follows that R 0 < λp,q ≤ for all w = (u, v) ∈ H.

+ |∇v|q )dx p q Ω (|u| + |v| )dx

ΩR(|∇u|

p

Ω


5

If w = (u, v) ∈ H is a weak nontrivial solution of problem (1.1) then multiplying the first equation of (1.1) by u, the second by v, and integrating by parts, we can get Z min{α1 , α2 } (|∇u|p + |∇v|q )dx Ω Z ≤ h1 (|∇u|p )|∇u|p + h2 (|∇v|q )|∇v|q dx Ω Z =λ Fu (x, u, v)u + Fv (x, u, v)v dx. Ω

On the other hand, using (H20 ) and the Young inequality, we have Z (Fu (x, u, v)u + Fv (x, u, v)v)dx Ω Z ≤ |Fu (x, u, v)||u| + |Fv (x, u, v)||v| dx Ω Z Z p(q−1) q(p−1) p−1 p |v|q−1 + |u| q |u|dx + C |v|dx |u| + |v| ≤C Ω

Ω

Z p Z q |u| |v|q |v| |u|p p q ≤ C (|u| + |v| )dx + C + p dx + C + q dx p q Ω Ω Ω p−1 q−1 Z ≤ C (|u|p + |v|q )dx. Z

Ω

Hence, Z

p

min{α1 , α2 }

q

(|∇u| + |∇v| )dx ≤ λC Ω

and taking λ∗ =

λ min{α1 , α2 } Cp,q

Z

(|u|p + |v|q )dx

Ω

> 0, we get the proof of Theorem 1.3.

In order to prove Theorem 1.4, we shall use critical point theory. For all λ ∈ R, we consider the functional Tλ : H → R given by Tλ (w) = J(w) − λI(w),

w ∈ H.

(2.1)

By (H2), a simple computation implies that Tλ is well-defined and of C 1 class in H. Thus, weak solutions of system (1.1) correspond to the critical points of Tλ . Lemma 2.1. The functional Tλ given by (2.1) is weakly lower semicontinuous. Proof. We first prove that J is weakly lower semicontinuous in H. Let w0 = (u0 , v0 ) ∈ H and > 0 be fixed. Using the properties of lower semicontinuous function (see [6, section I.3]), it is enough to prove that there exists δ > 0 such that J(w) ≥ J(w0 ) − ,

∀w = (u, v) ∈ H with kw − w0 kH < δ.

(2.2)

6


Using condition (H5) and H¨ older’s inequality we deduce there exists a positive constant C > 0 such that Z

|h1 (|∇u0 |p )||∇u0 |p−2 |∇u0 ||∇u − ∇u0 |dx Z |h2 (|∇v0 |q )||∇v0 |q−2 |∇v0 ||∇v − ∇v0 |dx − Ω Z Z p−1 ≥ J(w0 ) − β1 |∇v0 |q−1 |∇v − ∇v0 |dx |∇u0 | |∇u − ∇u0 |dx − β2

J(w) ≥ J(w0 ) −

Ω

Ω

Ω

≥ J(w0 ) − β1

Z

|∇u0 |p dx

p−1 Z p

Ω

− β2

|∇u − ∇u0 |p dx

1

p

Ω

q−1 Z 1 q q |∇v0 |q dx |∇v − ∇v0 |q dx

Z

Ω

Ω

= J(w0 ) − Ckw − w0 kH ,

∀(u, v) ∈ H,

where C is a positive constant. It is clear that taking δ =

C

relation (2.2) holds true for all

w = (u, v) ∈ H with kw − w0 kH < δ. Thus we proved that J is strongly lower semicontinuous. Taking into account the fact that J is convex then by [6, corollary III.8] we conclude that J is weakly lower semicontinuous in H. Next, we prove that the functional I given by (2.1) is weakly continuous in H. This follows that Tλ is weakly lower semicontinuous in H. Let {wm } = {(um , vm )} be a sequence converges weakly to w = (u, v) in H. We will show that Z lim

m→∞ Ω

Z F (x, um , vm )dx =

F (x, u, v)dx. Ω

Indeed, we have Z h Z i F (x, um , vm ) − F (x, u, v) dx = ∇F (x, w + θm (wm − w)) · (wm − w)dx Ω Ω Z = Fu (x, u + θ1,m (um − u), v + θ2,m (vm − v))(um − u)dx Ω Z + Fv (x, u + θ1,m (um − u), v + θ2,m (vm − v))(vm − v)dx, Ω

where θm = (θ1,m , θ2,m ) and 0 ≤ θ1,m (x), θ2,m (x) ≤ 1 for all x ∈ Ω.

(2.3)


7

Now, using (H2) and H¨ older’s inequality we conclude that Z [F (x, um , vm ) − F (x, u, v)]dx Ω Z ≤ |Fu (x, u + θ1,m (um − u), v + θ2,m (vm − v)||um − u|dx Ω Z |Fv (x, u + θ1,m (um − u), v + θ2,m (vm − v)||vm − v|dx + Ω Z q(p−1) ≤C 1 + |u + θ1,m (um − u)|p−1 + |v + θ2,m (vm − v)| p |um − u|dx Ω Z p(q−1) +C 1 + |u + θ1,m (um − u)| q + |v + θ2,m (vm − v)|q |vm − v|dx

(2.4)

Ω

≤ C |Ω|

p−1 p

q(p−1) p + ku + θ1,m (um − u)kp−1 Lp + kv + θ2,m (vm − v)kLq

kum − ukLp

q−1 p(q−1) q−1 kvm − vkLq , + C |Ω| q + ku + θ1,m (um − u)kLp q + kv + θ1,m (vm − v)kL q where |Ω| denotes the Lebesgue measure of Ω in RN . On the other hand, since H ,→ Lp (Ω) × Lq (Ω) is compact, the sequence {wm } converges to w = (u, v) in the space Lp (Ω) × Lq (Ω), i.e., {um } converges strongly to u in Lp (Ω) and {vm } converges strongly to v in Lq (Ω). Hence, it is easy to see that the sequences {ku + θ1,m (um − u)kLp (Ω) } and {kv + θ2,m (vm − v)kLq (Ω) } are bounded. Thus, it follows from (2.4) that relation (2.3) holds true. The proof of Lemma 2.1 is proved.

Lemma 2.2. The functional Tλ is coercive and bounded from below. Proof. Consider M as in (Γ). By (H2) and (H4), there exists Cλ > 0 such that for all (t, s) ∈ R2 and a.e. x ∈ Ω we deduce that λF (x, t, s) ≤

λp,q min{α1 , α2 } G2 (t, s) + Cλ . 2M max{p, q}

Hence, using (H5), Z Z Z α1 α2 p q |∇u| dx + |∇v| dx − λ F (x, u, v)dx Tλ (u) ≥ p Ω q Ω Ω Z Z Z λp,q min{α1 , α2 } α1 α2 ≥ |∇u|p dx + |∇v|q dx − G2 (u, v) − Cλ dx p Ω q Ω 2M max{p, q} Ω Z Z Z Z λp α1 λq α2 α1 α2 p q p ≥ |∇u| dx + |∇v| dx − |u| dx − |v|q dx − Cλ |Ω| p Ω q Ω 2p Ω 2q Ω α1 α2 = kukp1,p + kvkp1,q − Cλ |Ω|, 2p 2q so the functional Tλ is coercive and bounded from below.

(2.5)

(2.6)

Lemma 2.3. If w = (u, v) ∈ H is a weak solution of system (1.1) then u ≥ 0 and v ≥ 0 in Ω.

8


Proof. From (H1), if t < 0 then F (x, t, s) = F (x, t, s) = F (x, 0, s) for all t < 0, all s ∈ R, x ∈ Ω and thus, Ft (x, t, s) = lim t→t

F (x, t, s) − F (x, t, s) = 0 for x ∈ Ω. t−t

Similarly, if s < 0 then Fs (x, t, s) = 0 for x ∈ Ω and all t ∈ R. Now, if w = (u, v) is a weak solution of system (1.1), then we have 0 = hTλ0 (w), w− i Z = h1 (|∇u|p )|∇u|p−2 ∇u · ∇u− + h2 (|∇v|p )|∇v|q−2 ∇v · ∇v − dx Ω Z h i −λ Fu (x, u, v)u− + Fv (x, u, v)v − dx Ω Z Z − p ≥ α1 |∇u | dx + α2 |∇v − |q dx, Ω

Ω

where u− = min{u(x), 0}, v − = min{v(x), 0} are the negative parts of u and v, respectively. It follows that u(x) ≥ 0 and v(x) ≥ 0 for a.e. x ∈ Ω.

By Lemmas 2.1-2.3, applying the Minimum principle (see [17, p. 4, Theorem 1.2]), the functional Tλ has a global minimum and thus system (1.1) admits a nonnegative weak solution w1 = (u1 , v1 ) ∈ H. The following lemma shows that the solution w1 is not trivial provided λ is large enough. Lemma 2.4. There exists a constant λ∗ > 0 such that for all λ ≥ λ∗ , inf H Tλ < 0 and hence the solution w1 6≡ 0 . Proof. Indeed, let B 0 be a sufficiently large compact subset of B, where B is a ball such that F (x, t0 , s0 ) > 0 for all x ∈ B and some t0 , s0 > 0. Let a smooth function w0 = (u0 , v0 ) with compact support in B, such that u0 (x) = t0 , v0 (x) = s0 on B 0 , 0 ≤ u0 (x) ≤ t0 , 0 ≤ v0 (x) ≤ s0 on B\B 0 . Then we have Z

Z F (x, u0 , v0 )dx =

Ω

B0

Z F (x, u0 , v0 )dx +

Z ≥ B0

Z ≥ B0

B\B 0

F (x, u0 , v0 )dx

Z F (x, t0 , s0 )dx − C

B\B 0

(1 + |u0 |p + |v0 |q )dx

F (x, t0 , s0 )dx − C(1 + |t0 |p + |s0 |q )|B\B 0 | > 0,


9

provided that |B\B 0 | > 0 is small enough. So, we deduce that Z Z Z β1 β2 |∇u0 |p dx + |∇v0 |p dx − λ F (x, u0 , v0 )dx p Ω q Ω Ω Z Z β β1 2 |∇u0 |p dx + |∇v0 |p dx ≤ p Ω q Ω Z F (x, t0 , s0 )dx − C(1 + |t0 |p + |s0 |q )|B\B 0 | . −λ

Tλ (u0 , v0 ) ≤

B0

Hence, if B 0 is large enough, there exists λ∗ such that for all λ ≥ λ∗ we have Tλ (w0 ) < 0, thus w1 6≡ 0. Moreover Tλ (w1 ) < 0 for all λ ≥ λ∗ .

Our idea is to obtain the second weak solution w2 = (u2 , v2 ) ∈ H by applying the mountain pass theorem in [2]. To this purpose, we first show that for all λ ≥ λ∗ , the functional Tλ has the geometry of the mountain pass theorem. Lemma 2.5. There exist a constant ρ ∈ (0, kw1 kH ) and a constant r > 0 such that Tλ (w) ≥ r for all w ∈ H with kwkH = ρ. Proof. By (H2) and (H3), we have for all (t, s) ∈ R2 and all x ∈ Ω, λF (x, t, s) ≤

λp,q min{α1 , α2 } G1 (t, s) + Cλ (|t|α + |s|β ), 2M max{p, q}

(2.7)

where p < α < p∗ , q < β < q ∗ . Hence, using the continuous embeddings, we get Z Z λp,q min{α1 , α2 } α2 q |∇u| dx + |∇v| dx − G1 (u, v)dx q Ω 2M max{p, q} Ω Ω Z − Cλ (|u|α + |v|β )dx Ω Z Z Z λp α1 α1 α2 p q ≥ |∇u| dx + |∇v| dx − |u|p dx p Ω q Ω 2p Ω Z Z λq α2 − |u|q dx − Cλ (|u|α + |v|β )dx 2q Ω Ω α1 α2 p p ≥ kuk1,p + kvk1,q − Cλ kukα1,p − Cλ kvkβ1,q 2p 2q α α 1 2 = − Cλ kukα−p kukp1,p + − Cλ kvkβ−q kvkq1,q . 2p 2q

α1 Tλ (u) ≥ p

Z

p

Since p < α < p∗ , q < β < q ∗ , there are positive constants ρ < kw1 kH and r such that Tλ (w) ≥ r for all w ∈ H with kwkH = ρ. Lemma 2.6. The functional Tλ satisfies the Palais-Smale condition in H.

10


Proof. By Lemma 2.2, we deduce that Tλ is coercive on H. Let {wm } = {(um , vm )} be a Palais-Smale sequence for the functional Tλ in H, i.e. Tλ0 (um ) → 0 in H −1 as m → ∞,

|Tλ (um )| 5 c for all m,

(2.8)

where H −1 is the dual space of H. Since Tλ is coercive on H, relation (2.8) implies that the sequence {wm } is bounded in H. Since H is reflexive, there exists w = (u, v) ∈ H such that, passing to a subsequence, still denoted by {wm }, it converges weakly to w in H. Hence, {kwm − wkH } is bounded. This and (2.8) imply that Tλ0 (wm )(wm − w) converges to 0 as m → ∞. Using the condition (H1) combined with H¨ older’s inequality we conclude that Z Z q(p−1) |um − u|dx |Fu (x, um , vm )||um − u|dx ≤ C 1 + |um |p−1 + |vm | p Ω

Ω

≤ C |Ω|

p−1 p

+

kum kp−1 Lp

q(p−1) p Lq

+ kvm k

(2.9)

kum − ukLp

and Z

Z |Fv (x, um , vm )||vm − v|dx ≤ C

1 + |um |

p(q−1) q

+ |vm |q−1 |vm − v|dx

Ω

Ω

≤ C |Ω|

q−1 q

p(q−1) q Lp

+ kum k

(2.10) +

kvm kq−1 Lq

kvm − vkLq .

It follows from relations (2.9) and (2.10) that lim

m→∞

I 0 (wm ), wm − w = 0.

Combining this with (2.8) and the fact that

0

J (wm ), wm − w = Tλ0 (wm ), wm − w + λ I 0 (wm ), wm − w imply that Z

lim J 0 (wm ), wm − w = lim

m→∞

m→∞ Ω

h1 (|∇um |p )|∇um |p−2 ∇um · (∇um − ∇u)dx

Z + lim

m→∞ Ω

h2 (|∇vm |q )|∇vm |q−2 ∇vm · (∇vm − ∇v)dx = 0. (2.11)

On the other hand, since {wm } converges weakly to w in H, we get lim

m→∞

0

Z

J (w), wm − w = lim

h1 (|∇u|p )|∇u|p−2 ∇u · (∇um − ∇u)dx Z + lim h2 (|∇v|q )|∇v|q−2 ∇v · (∇vm − ∇v)dx = 0. (2.12)

m→∞ Ω

m→∞ Ω


11

Using the condition (H6), relations (2.11) and (2.12) imply that γ1 kum − ukp1,p + γ1 kvm − vkq1,q Z ≤ h1 (|∇um |p )|∇um |p−2 ∇um − h1 (|∇u|p )|∇u|p−2 ∇u · (∇um − ∇u)dx Ω Z + h2 (|∇vm |q )|∇vm |q−2 ∇vm − h2 (|∇v|q )|∇v|q−2 ∇v · (∇vm − ∇v)dx

(2.13)

Ω

= J 0 (wm ) − J 0 (w), wm − w , which approaches 0 as m → ∞. Therefore, {wm } converges strongly to w in H and the functional Tλ satisfies the Palais-Smale condition in H.

Proof of Theorem 1.4. By Lemmas 2.1-2.4, system (1.1) admits a non-negative, non-trivial weak solution w1 = (u1 , v1 ) as the global minimizer of Tλ . Setting c := inf

max

χ∈Γ w∈χ([0,1])

Tλ (w),

(2.14)

where Γ := {χ ∈ C([0, 1], H) : χ(0) = 0, χ(1) = w1 }. Lemmas 2.5, 2.6 show that all assumptions of the mountain pass theorem in [2] are satisfied, Tλ (w1 ) < 0 and kw1 kH > ρ. Then, c is a critical value of Tλ , i.e. there exists w2 = (u2 , v2 ) ∈ H such that Tλ0 (w2 )(ϕ) = 0 for all ϕ ∈ H or w2 is a weak solution of (1.1). Moreover, w2 is not trivial and w2 6≡ w1 since Tλ (w2 ) = c > 0 > Tλ (w1 ). Theorem 1.4 is completely proved.

3. Appendix In this section, we prove that the function h : [0, +∞) → R defined by h(t) = 1 + √

t , 1 + t2

t≥0

satisfies the condition

h(|ξ|p )|ξ|p−2 ξ − h(|η|p )|η|p−2 η · (ξ − η) ≥ γ0 |ξ − η|p ,

∀ξ, η ∈ RN ,

p ≥ 2,

(3.1)

where γ0 is a positive constant. Indeed, we are motivated by the proof of the following well-known inequality (see for example [16, Lemma A.0.5])

|ξ|

p−2

p−2

ξ − |η|

η · (ξ − η) ≥ Cp |ξ − η|p ,

where Cp is a positive constant depending on p.

∀ξ, η ∈ RN ,

p ≥ 2,

(3.2)

12


By homogeneity, we can assume that |ξ| = 1 and |η| ≤ 1. Moreover, by choosing a convenient basic in RN , we can assume that ξ = (1, 0, ..., 0),

η = (η1 , η2 , 0, ..., 0), and

q

η12 + η22 ≤ 1.

Then (3.1) is equivalent to the following inequality p p p−2 p−2 (η12 +η22 ) 2 (η12 +η22 ) 2 1 2 2 √ √ √ 2 (η + η ) η (1 − η ) + 1 + (η12 + η22 ) 2 η22 1+ 2 − 1+ 1 1 1 2 2 2 2 2 1+(η1 +η2 )p 1+(η1 +η2 )p ≥ γ0 . p (1 − η1 )2 + η22 2 Denote t =

|η| |ξ|

and s =

hξ,ηi |ξ||η| ,

1+ γ(t, s) =

√1 2

then we must show that the function −s

1+

√1 2

t+ 1+

p √ t 1+t2p

1 − 2ts + t2

tp−1 + 1 +

p

√ t 1+t2p

tp

p 2

is bounded from below. It is clear that 0 ≤ t ≤ 1 and −1 ≤ s ≤ 1. Direct calculation shows that fixed t,

∂γ ∂s

= 0 if tp p 1 tp p−1 1 + 1 + 1+ √ −s 1+ √ t+ 1+ t t 1 + t2p 1 + t2p 2 2 p−2 tp 1 + √12 + 1 + √1+t t 2p = (1 − 2ts + t2 ), p

then for the critical s = s∗ for f we have 1+

∗

γ(t, s ) =

+ 1+

p √ t 1+t2p

tp−2 .

p 1+

≥

√1 2

√1 2

p √ t 1+t2p 1)p−2

+ 1+ p(t +

1+ ≥ min t∈[0,1]

√1 2

(1 − 2ts∗ + t2 )

p−2 2

tp−2

p √ t 1+t2p 1)p−2

+ 1+ p(t +

1

tp−2 = γ0 > 0.

This completes the proof. Acknowledgments. The authors would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript.

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G.A. Afrouzi, Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran E-mail address: [email protected]

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N.T. Chung, Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam E-mail address: [email protected]

Z. Naghizadeh, Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran E-mail address: [email protected]