HOMOCLINIC SOLUTIONS FOR A PRESCRIBED MEAN CURVATURE RAYLEIGH p-LAPLACIAN EQUATION WITH A DEVIATING ARGUMENT

Abstract

In this paper, the prescribed mean curvature Rayleigh p-Laplacian equation with a deviating argument

1. Introduction

In resent years, The existence of homoclinic solutions have been studied widely, especially for the Hamiltonian systems and the p-Laplacian systems(see [1-4]). For example, in [1], Lzydorek, M and Janczewska, J studied the homoclinic solutions for a class of the second order Hamiltonian systems as the following form

where q ∈ Rn and V ∈ C1(R × Rn,R), V (t, q) = −K(t, q) + W(t, q) is T-periodic in t. And in [4], Lu, SP studied the homoclinic solutions for a class of second-order p-Laplacian differential systems with delay of the form

Nowadays, the prescribed mean curvature equation and its modified forms, which arises from some problems associated with differential geometry and physics such as combustible gas dynamics [5-7] have been studied widely. As researchers continue to study the prescribed mean curvature equation, the existence of the periodic solutions for the prescribed curvature mean equation attracts researchers’ attention and there are many papers about the existence of the periodic solutions for the prescribed curvature mean equation. For example, in [11], Feng discussed the existence of periodic solutions of a delay prescribed mean curvature Li´enard equation of the form

and in [12], Jin Li discussed the existence of periodic solutions for a prescribed mean curvature Rayleigh equation of the form

As is well known, a solution u(t) of Eq.(1.1) is named homoclinic (to 0) if u(t) → 0 and u'(t) → 0 as |t| → +∞. In addition, if u ≠ 0, then u is called a nontrivial homoclinic solution.

In [13], Liang and Lu studied the homoclinic solution for the prescribed mean curvature Duffing-type equation of the form

where f ∈ C1(R,R), p ∈ C(R,R), c > 0 is a given constant.

Recently, in [14], Wang studied the periodic solution for the following prescribed mean curvature Rayleigh equation with a deviating argument of the form:

where p > 1 and φp : R → R is given by φp(s) = |s|p−2s for s ≠ 0 and φp(0) = 0, g ∈ C(R2,R), e, τ ∈ C(R,R), g(t + ω, x) = g(t, x), f(t + ω, x) = f(t, x), f(t, 0) = 0, e(t + ω) = e(t) and τ (t + ω) = τ (t). Under the assumptions:

and

where a, r ≥ 1; m1 and m2 are positive constants. Through the transformation, (1) is equivalent to the system

By using Mawhin’s continuation theorem and given some sufficient conditions, the authors obtained that Eq.(1) has at least one periodic solution.

However, to the best of our knowledge, there are no papers about the studying of the homoclinic solutions for the prescribed mean curvature Rayleigh p-Laplacian equation. In order to solve this problem, in this paper, we consider the following the prescribed mean curvature Rayleigh p-Laplacian equation with a deviating argument

where p > 1 and φp : R → R is given by φp(s) = |s|p−2s for s ≠ 0 and φp(0) = 0, f ∈ C(R,R), g ∈ C(R2,R), g is T-periodic in the first argument. e(t), τ (t) are continuous T-periodic function and T > 0 is a given constant.

In order to study the homoclinic solution for Eq.(3), firstly, like in the work of Lzydorek and Janczewska in [1], Rabinowitz in [2], X. H. Tang and Li Xiao in [3] and Lu in [4], the existence of a homoclinic solution for Eq.(3) is obtained as a limit of a certain sequence of 2kT-periodic solutions for the following equation:

where k ∈ N, ek : R → R is a 2kT-periodic function such that

where ε0 ∈ (0, T) is a constant independent of k. In our approach,the existence of 2kT-periodic solutions to Eq.(4) is obtained by applying Mawhin’s continuation theorem [16].

The structure of the rest of this paper is as follows: Section 2, we state some necessary definitions and lemmas. Section 3, we prove the main result.

 

2. Preliminary

Throughout this paper, | · | will denote the absolute value and the Euclidean norm on R. For each k ∈ N, let C2kT = {u|u ∈ C(R,R), u(t + 2kT) = u(t)}, If the norms of are defined by respectively, then are all Banach spaces. Furthermore, for ϕ ∈ C2kT, , where r ∈ (1,+∞).

In order to use Mawhin’s continuation theorem, we first recall it.

Let X and Y be two Banach spaces, a linear operator L : D(L) ⊂ X → Y is said to be a Fredholm operator of index zero provided that

(a) ImL is a closed subset of Y,

(b) dimKerL = codimImL < ∞.

Let X and Y be two Banach spaces, Ω ⊂ X be an open and bounded set, and L : D(L) ⊂ X → Y is a Fredholm operator of index zero, and continuous operator N : Ω ⊂ X → Y is said to be L-compact in provided that

(c) is a relative compact set of X,

(d) is a bounded set of Y,

where we denote X1 = KerL, Y2 = ImL, then we have the decompositions X = X1 ⊕ X2, Y = Y1 ⊕ Y2, let P : X → X1, Q : Y → Y1 are continuous linear projectors(meaning P2 = P and Q2 = Q), and

Lemma 2.1 (16). Let X and Y be two real Banach spaces, and Ω is an open and bounded set of X, and L : D(L) ⊂ X → Y is a Fredholm operator of index zero and the operator is said to be L-compact in . In addition, if the following conditions hold:

(h1) Lx ≠ λNx, ∀(x, λ) ∈ ∂Ω × (0, 1);

(h2) QNx ≠ 0, ∀x ∈ KerL ∩ ∂Ω;

(h3) deg{JQN,Ω∩KerL, 0} ≠ 0, where J : ImQ → KerL is a homeomorphism, then Lx = Nx has at least one solution in

Lemma 2.2 ([4]). Let s ∈ C(R,R) with s(t+ω) ≡ s(t) and s(t) ∈ [0, ω], ∀t ∈ R. Suppose p ∈ (1,+∞), and u ∈ C1(R,R) with u(t + ω) = u(t). Then

Lemma 2.3. If u : R → R is continuously differentiable on R, a > 0, μ > 1 and p > 1 are constants, then for every t ∈ R, the following inequality holds

In order to study the existence of 2kT-periodic solutions for Eq.(1.2), for each k ∈ N, from (1.3) we observe that ek ∈ C2kT.

Lemma 2.4 ([18]). Suppose τ ∈ C1(R,R) with τ (t + ω) ≡ τ (t) and τ' (t) < 1, ∀t ∈ [0, ω]. Then the function t−τ (t) has an inverse μ(t) satisfying μ ∈ C(R,R) with μ(t + ω) ≡ μ(t) + ω, ∀t ∈ [0, ω].

Throughout this paper, besides τ being a periodic function with period T, we suppose in addition that τ ∈ C1(R,R) with τ' (t) < 1, ∀t ∈ [0, T].

Remark 2.1. From the above assumption, one can find from Lemma 2.4 that the function (t−τ (t)) has an inverse denoted by μ(t). Define Clearly, σ0 ≥ 0 and 0 ≤ σ1 < 1.

Lemma 2.5 ([3]). Let be a 2kT-periodic function for each k ∈ N with

where A0, A1 and A2 are constants independent of k ∈ N. Then there exists a function u ∈ C1(R,Rn) such that for each interval [c, d] ⊂ R, there is a subsequence {ukj} of {uk}k∈N with uniformly on [c, d].

The system (4) is equivalent to the system

where

Let Xk = {ω = (u(t), v(t))⊤ ∈ C(R,R2), ω(t) = ω(t + 2kT)} and Yk = {ω = (u(t), v(t))⊤ ∈ C(R,R2), ω(t) = ω(t + 2kT)}, where the norm ||ω|| = max{|u|0, |v|0} with It is obvious that Xk and Yk are Banach spaces.

Now we define the operator

where D(L) = {ω|ω = (u(t), v(t))⊤ ∈ C1(R,R2), ω(t) = ω(t + 2kT)}.

Let Zk = {ω|ω = (u(t), v(t))⊤ ∈ C1(R,R × Bk), ω(t) = ω(t + 2kT)}, where Bk = {x ∈ R, |x| < 1, x(t) = x(t + 2kT)}. Define a nonlinear operator as follows:

where and Ω is an open and bounded set. Then problem (6) can be written as

we know

then ∀t ∈ R we have u′(t) = 0, v′(t) = 0, obviously u = a1 ∈ R, v = a2 ∈ R, thus KerL = R2, and it is also easy to prove that Therefore, L is a Fredholm operator of index zero.

Let

Let then it is easy to see that:

where

For all such that we have is a relative compact set of Xk, is a bounded set of Yk, so the operator N is L-compact in .

For the sake of convenience, we list the following assumption which will be used by us in studying the existence of homoclince solutions to the Eq.(3) in Section 3.

[H1] There exists constants α and β > 0 such that

[H2] There exists constants m0 and m1 > 0 such that

[H3] e ∈ C(R,R) is a bounded function with e(t) ≠ 0 and

Remark 2.2. From (5), we can see that So if [H3] holds, then for each

 

3. Main results

In order to study the existence of 2kT-periodic solutions to system (6), we firstly study some properties of all possible 2kT-periodic solutions to the following system:

where (uk, vk)⊤ ∈ Zk ⊂ Xk. For each k ∈ N and all λ ∈ (0, 1], let Δ represent the set of all the 2kT-periodic solutions to the above system.

Theorem 3.1. Assume that conditions [H1]-[H3] hold,

and there exists a positive constant d0 such that

where

then for each k ∈ N, if (u, v)⊤ ∈ Δ, there are positive constants ρ1, ρ2, ρ3 and ρ4 which are independent of k and λ, such that

Proof. For each k ∈ N, if (u, v)⊤ ∈ Δ, it must satisfy the system (7). Multiplying the second equation of (7) by u′(t) and integrating from −kT to kT, we have

In view of [H1] and [H2] and by Hölder inequality, we get

Furthermore,

and by Lemma 2.4,

It follows from Remark 2.1 that

Substituting (9) into (8) and combining with Remark 2.2, we can obtain

which yields

Multiplying the second equation of (7) by u(t) and integrating from −kT to kT, we have

From the equality above, we have

Since and in view of [H1], [H2] and Lemma 2.2, we can get

By applying (9) to (11), we have

From the inequality above, we can see that

and

Substituting (10) into (13), we get

Since it is easy to see that there exists a constant d0 such that

Substituting (14) into (10), we obtain

It follows from Lemma 2.2 that

In view of (14) and (15), we have

then we get

Clearly, ρ1 is independent of k and λ. Furthermore, substituting (14) and (15) into (12), we can see that

Multiplying the second equation of (7) by v′(t) and integrating from −kT to kT, we have

From the first equation of (7), we can see that thus

Substituting (19) into (18) and in view of [H2], we get

It follows from (14) and (16) that

Applying the Lemma 2.2 again, we have

then combining (17) and (20) gives

It follows from that

Clearly, ρ2 is independent of k and λ.

Clearly, ρ3 is independent of k and λ. Let define then from the second equation of (7), we can obtain

and also ρ4 is independent of k and λ. Therefore, From (16), (22), (23) and (24), we know ρ1, ρ2, ρ3 and ρ4 are constants independent of k and λ. Hence the conclusion of Theorem 3.1 holds. □

Theorem 3.2. Assume that the conditions of Theorem 3.1 are satisfied . Then, for each k ∈ N, system (7) has at least one 2kT-periodic solution (uk(t), vk(t))⊤ in Δ ⊂ Xk such that

whereρ1, ρ2, ρ3 and ρ4 are constants defined by Theorem 3.1.

Proof. In order to use Lemma 2.1, for each k ∈ N, we consider the following system:

where Let Ω1 ⊂ Xk represent the set of all the 2kT-periodic solutions of system (25). Since (0, 1) ⊂ (0, 1], then Ω1 ⊂ Δ, where Δ is defined by Theorem 3.1. If (u, v)⊤ ∈ Ω1, by using Theorem 3.1, we have

Let Ω2 = {ω = (u, v)⊤ ∈ KerL,QNω = 0}, if (u, v)⊤ ∈ Ω2, then (u, v)⊤ = (a1, a2)⊤ ∈ R2(constant vector) and we can see that

i.e.,

Multiplying the second equation of (26) by a1 and combining with [H2], we have

Thus

Now, if we define it is easy to see that Ω1 ∪ Ω2 ⊂ Ω. So, condition (h1) and condition (h2) of Lemma 2.1 are satisfied. In order to verify the condition (h3) of Lemma 2.1, let

where J : ImQ → KerL is a linear isomorphism, J(u, v) = (v, u)⊤. From assumption [H1] and [H2], we have

Hence,

So, the condition (h3) of Lemma 2.1 is satisfied. Therefore, by using Lemma 2.1, we see that Eq.(6) has a 2kT-periodic solution (uk, vk)⊤ ∈ Ω. Obviously, (uk, vk)⊤ is a 2kT-periodic solution to Eq.(2) for the case of λ = 1, so (uk, vk)⊤ ∈ Δ. Thus, by using Theorem 3.1, we have

Hence the conclusion of Theorem 3.2 holds. □

Theorem 3.3. Suppose that the conditions in Theorem 3.1 hold, then Eq.(1) has a nontrivial homoclinic solution.

Proof. From Theorem 3.2, we see that for each k ∈ N, there exists a 2kT-periodic solution (uk, vk)⊤ to Eq.(2) with

where ρ1, ρ2, ρ3, ρ4 are constants independent of k ∈ N. And uk(t) is a solution of (2), so

with implies that vk(t) is continuously differentiable for t ∈ R. Also, from (27), we have |vk|0 ≤ ρ2 < 1. It follows that is continuously differentiable for t ∈ R, i.e.,

By using (27) again and combining with φq(s) = |s|q−2s for s ≠ 0, then we have

Clearly, ρ5 is a constant independent of k ∈ N. From Lemma 2.5, we can see that there is a function u0 ∈ C1(R,Rn) such that for each interval [a, b] ⊂ R, there is a subsequence {ukj} of {uN}k∈N with uniformly on [a, b]. In the following, we show that u0(t) is just a homoclinic solution to Eq.(4).

For all a, b ∈ R with a < b, there must be a positive integer j such that for j > j0, [−kjT, kjT − ε0] ⊂ [a − α, b + α]. So, for j > j0, from (3) and (26) we see that

Then from (29) we can have

Since uniformly for t ∈ [a, b] and is continuous differentiable for t ∈ [a, b], we can have

Considering that a, b are two arbitrary constants with a < b, it is easy to see that u0(t), t ∈ R is a solution to system (1).

Now, we prove u0(t) → 0 and u'(t) → 0 as |t| → ∞.

Since

Clearly, for every i ∈ N if kj > i, then by (14) and (15), we have

Let i → +∞, j → +∞, we have

and then

as r → +∞. So, by using Lemma 2.3 as |t| → +∞, we obtain

Finally, we will proof

From (27), we know

Then, we have

If (32) does not hold, then there exist and a sequence {tk} such that

and

Then, for we can have

It follows that

which contradicts (30), thus (32) holds. Clearly, u0(t) ≠ 0, otherwise e(t) ≡ 0, which contradicts assumption (H3). Hence the conclusion of Theorem 3.3 holds. □