Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian

In this paper we prove the existence and multiplicity of subharmonic solutions for nonlinear equations involving the singular \begin{document}$ \phi $\end{document} -Laplacian. Such equations are in particular motivated by the one-dimensional mean curvature problems and by the acceleration of a relativistic particle of mass one at rest moving on a straight line. Our approach is based on phase-plane analysis and an application of the Poincare-Birkhoff twist theorem.

1. Introduction. In this paper we deal with the existence and multiplicity of subharmonic solutions of nonlinear second order equation (φ(x )) + g(t, x) = 0 (1) involving the singular φ-Laplacian, where φ : (−a, a) → R is an increasing homeomorphism with φ(0) = 0, and g : R × R → R is a continuous and periodic function with period 2π with respect to time t. Also, we assume that, there are constants ε 0 > 0 and d 0 > 0, such that g(t, x) satisfies the following sign condition: x) ≥ ε 0 , ∀t ∈ [0, 2π] and |x| ≥ d 0 .
In case φ is the identity operator, Eq. (1) is the classical second order forced Duffing equation x + g(t, x) = 0.
(2) The existence and multiplicity of subharmonic solutions have been studied by many researchers under various growth conditions on the function g. For example, see [29,10,11,12,15]. Also, some extensions can be found in [16,22,23].
In case φ is an increasing homeomorphism from (−a, a) to R, the existence of solutions of various boundary value problems have been studied by Bereanu and Mawhin based on the method of upper and lower solutions and Leray-Schauder degree, see [2,3,4,5]. We also note the work of Bereanu and Torres on the multiple existence of periodic solutions of relativistic forced pendulum by using critical point theory [6].
However, there are few results on the multiplicity of subharmonic solutions of Eq. (1). Recently, Boscaggin, Garrione and Feltrin, Fonda and Toader, Donde and Zanolin applied the Poincaré-Birkhoff twist theorem to discuss related problems, see [8,7,19,14]. This paper is another result in this direction. Using the careful phaseplane analysis based on the sign condition (G 0 ), we apply the Poincaré-Birkhoff twist theorem to prove the following multiplicity result. Theorem 1.1. Assume that φ : (−a, a) → R is an increasing homeomorphism with φ(0) = 0, and g(t, x) is a continuous function and 2π-periodic in t, such that solutions of Eq. (1) are unique with respect to initial value. If ( G 0 ) holds, then Eq. (1) has at least one 2π-periodic solution. Moreover, for any prime number n, there is m 0 (n) > 0 such that, for any positive integer m ≥ m 0 (n), Eq. (1) has at least two subharmonic solutions x n,m,k (t), k = 1, 2, with minimal period 2mπ. Furthermore, Some parts of our arguments are motivated by Ding and Zanolin [12]. In [12], they proved the multiplicity of subharmonic solutions for sublinear Duffing equation (2) via the Poincaré-Birkhoff twist theorem. To apply the Poincaré-Birkhoff twist theorem, one needs to construct an annulus such that the solution z(t; z 0 ) = (x(t; x 0 , y 0 ), y(t; x 0 , y 0 )) satisfies that arg(z(T ; z 0 )) − arg(z 0 ) + 2nπ < 0 (or > 0), z 0 ∈ Γ − , arg(z(T ; z 0 )) − arg(z 0 ) + 2nπ > 0 (or < 0), z 0 ∈ Γ + , where Γ − and Γ + are the inner and outer boundary of the annulus, respectively. The difficulty of constructing such annulus for sublinear Duffing equation (2) is how to find the inner boundary Γ − . Sometimes, the solutions may pass through the origin. In this case we can not compute and estimate the argument because the argument function arg(z) is not well-defined at z = 0. A good idea from Ding and Zanolin can be used to overcome this difficulty. They introduced a transformation y(t) = x(t)−x 0 (t) with x 0 (t) a 2π-periodic solution of Eq. (2) (obtained by a simple application of topological method since there is a natural priori bound for sublinear equations). Then Eq. (2) is equivalent to Each nonzero solution of Eq. (3) does not pass through the origin by uniqueness of solutions. Thus one can compute and estimate the argument arg(z(t)) for all t ∈ R. But in our case, φ is not linear. The method introduced above is not easy to use. Fortunately, we can prove that the solutions of (1) have spiral properties, that is there are two spiral curves guiding the solutions of (1) in the phase plane, and forcing them to rotate around the origin as they increase in norm. Then each solution of (1) with fixed number of zeros starting from outside of a large disk in the phase plane does not pass through the origin. This property and the angular feature of the fixed points obtained by Pincaré-Birkhoff twist theorem inspire us to consider a modified equation of (1) instead. We mention here that in recent, Fonda and Sfecci [17] developed the so-called admissible spiral method, a tool introduced by the same authors in [18], to prove the existence of infinitely many periodic solutions for weakly coupled superlinear second order systems. The argument for the spiral property is also used to obtain the existence of infinitely many periodic solutions for superlinear second order equation (2) without the sign condition (G 0 ) [32]. The rest of paper is organized as follows. For special φ(x) = x √ 1−x 2 , in Section 2, we give an application of Theorem 1.1 in nonlinear systems under relativistic effect. In Section 3, we will prove some basic geometric properties of the solutions of the modified system. In particular, we will prove that the solutions of the modified system have spiral properties, that is, there are two spiral curves guiding the solutions in the phase plane, and forcing them to rotate around the origin as they increase in norm. Finally, in Section 4, we will give the proof of Theorem 1.1 using the Poincaré-Birkhoff twist theorem.
2. Subharmonic mechanical vibrations of nonlinear systems under relativistic effect. Relativistic oscillator models for particle motions are a subject of rather broad interest nowadays, which have been widely used in different branches of theoretical physics such as quantum mechanics, statistical mechanics, superconductivity theory, nuclear physics, and so forth [21,27,1].
Consider the equation describing the relativistic motion of molecule that interacts with another molecule d dt where the restoring term g(x) derives from one of the one-dimensional potentials and the forced term p(t) is 2π-periodic and continuous. Here, m 0 is the rest unit mass of the particle and c is the light speed. We normalize the equation by restricting the light speed c = 1. Some recent papers [24,25,26], in this field, have taken into account chaos, the relativistic dependence between the oscillator energy and the order of the formed resonances with various special potentials.
In case there is no forced term, Eq. (4) is the autonomous system Define the energy function by H(x, y) holds, then for large enough h, all the solutions with the orbit Γ h = {(x, y) : H(x, y) = h} (c ± , 0) are periodic with the fundamental period For various potentials, numerical results on the relations between the fundamental period T h and "energy" h (see Fig. 1) have shown that T h can be arbitrarily large, if h is large enough, that is, Eq. (5) has a natural property of second order sublinear system. Thus, if then it follows that there is n * > 0 such that, for any integer n ≥ n * , Eq. (5) has at least one subharmonic solution with minimal period 2nπ. As a direct application of Theorem 1.1, we have the same conclusion as the autonomous system for the periodically perturbed equation (4).
is a continuous function such that solutions of Eq. (4) are unique with respect to initial value, and p(t) is 2π-periodic and continuous. If ( G 1 ) holds, then Eq. (4) has at least one 2π-periodic solution and, there is N * > 0 such that, for any positive integer N > N * , Eq. (1) has at least one subharmonic solution x N (t) with minimal period 2N π. Moreover, 3. Geometric properties of the solutions of a modified system. Consider the equivalent system of Eq. (1) of the form We assume in what follows that g(t, x) is continuous and (G 0 ) holds. Moreover, we assume, without loss of generality, that every solution of system (6) is unique with respect to initial value.
Remark 3.1. If the solution of system (6) is not unique with respect to initial value, we can get same result by using an approximation approach as in [11].
For any given constants R 0 , R 1 with R 1 > R 0 > 0, let K(·) ∈ C ∞ be a function such that We consider the following modified system with the Hamiltonian In this section we will prove some basic geometric properties of the solutions of (7). In particular, we will prove that solutions of (7) have spiral properties, that is there are two spiral curves guiding the solutions of (7) in the phase plane and forcing them to rotate around the origin as they increase in norm.
Lemma 3.1. Every solution of the Cauchy problem associated with system (7) is defined on the whole t-axis.
Thus the solution (x(t; x 0 , y 0 ), y(t; x 0 , y 0 )) can not go to infinity for |t| ≤ T . Therefore the global existence of solutions is proved.
Note that the uniqueness of the solution with respect to initial value implies the continuity of the solution with respect to initial value combined with the global existence of solutions we have the following lemma Lemma 3.2. The solutions of (7) have elastic property, that is, for any given positive constants T > 0 and b 0 > 0, there is r b0 > 0 such that the inequality |(x 0 , y 0 )| ≥ r b0 implies |(x(t; x 0 , y 0 ), y(t; x 0 , y 0 ))| ≥ b 0 , for |t| ≤ T .
Furthermore, we prove that the solutions of system (7) have slow oscillatory properties.

XIYING SUN, QIHUAI LIU, DINGBIAN QIAN AND NA ZHAO
We will describe the behaviour of the solution in the following steps.
Step 1. Without loss of generality, let z(t) start from the line x = −R 2 and choose y 0 ≥ R 2 . For |x(t)| ≤ R 2 and y(t) ≥ R 2 we have Using x (t) = φ −1 (y(t)) > 0, we get Then for r 0 ≥ c 1 + 3R 2 we have Step g(t, s)ds, respectively. By the fact that φ is an increasing homeomorphism onto R and the condition ( G 0 ), we have Moreover, are two convex curves connecting the line x = R 2 and the x-axis in D 2 . Let h(t) = H(x(t), y(t)) and h(t) = H(x(t), y(t)). It is easy to verify that h (t) ≥ 0 and h (t) ≤ 0, for r(t) ≥ R 2 and z(t) ∈ D 2 .
On the other hand, from Lemma 3.3, θ (t) < 0 in Thus, there is c 2 > 0 such that θ (t) < −c 2 for z(t) ∈ E 2 which implies z(t) will escapes from E 2 at some time t 2 . Hence, we can assume that t 2 = s 2 , that is y(s 2 ) = 0 and x(s 2 ) = r(s 2 ).
Finally, for given positive integer N , define Take R (N ) 3 = ζ −N (R 2 ), then ζ N (r), ζ −N (r) are the functions such that the conclusion of the lemma holds. The proof is thus completed.
4. Existence of infinitely many subharmonic solutions. In this section we will give the proof of Theorem 1.1 by using the Poincaré-Birkhoff twist theorem.
For any prime number n, we will prove that there is m 0 (n) > 0, such that for any positive integer m ≥ m 0 (n) > 0, there exists an annulus A n,m such that the Poincaré-Birkhoff twist theorem can be used on A n,m . Firstly, we find the inner boundary of A n,m . According to Lemma 3.5, for given R ). Denote the annulus by ). Note that from the polar form of the system (7), θ (t) = Φ(t, θ(t), r(t)), where Φ(t, θ, r) is a continuous function, 2π-periodic with respect to t and θ. So Φ(t, θ, r) has negative lower and upper bounds on compact annulus A(R If there exists t 1 ∈ (0, 2mπ) such that r(t 1 ) < R (n) 3 or r(t 1 ) > R (n) 5 , according to Lemma 3.5, we know that θ(t 1 ) − θ(0) < −2(n + 1)π.
With the arguments discussed above, P m is an area-preserving homeomorphism with boundary twisting on the annulus A n,m .
Besides, since the 2π-periodic system (6) admits a positively bounded solution and it's Poincaré map is defined on R 2 , we can use the Massera theorem (see Massera [28] or Theorem 4.8 and Corollary 4.3 in [9]) to prove that Eq. (1) has at least one 2π−periodic solution.
Moreover, for fixed n and k, if there are a sequence {m l } and a bounded annulus with R * < +∞, such that z n,m l ,k (t) ∈ A * , for t ∈ [0, 2m l π].