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Periodic solutions of p-Laplacian equations via rotation numbers

We investigate the existence and multiplicity of periodic solutions of the p-Laplacian equation (φp(x′)) ′ + f(t, x) = 0. Both asymptotically linear and partially superlinear nonlinearities are studied, in absence of global existence and uniqueness conditions on the solutions of the associated Cauchy problems and the sign assumption on f . We use a approach of rotation number in the p-polar coordinates transformation, together with the phase-plane analysis of the rotational properties of large solutions and a recent version of Poincaré-Birkhoff theorem for Hamiltonian systems, for obtaining multiplicity results of p-Laplacian equation in terms of the gap between the rotation numbers of referred piecewise p-linear systems at zero and infinity.

These p-Laplacian equations are the generalization of the second order differential equations where p = 2. It is not restrictive to assume that the period of f in t is 2π p . When f is T -periodic in t, we can use a similar discussions on replacing 2π p -period with T -period to obtain the existence and multiplicity of T -periodic solutions of the p-Laplacian equation (1.1). For the results corresponding to the existence of periodic solutions of (1.1) based on topological degree theory and variational method one can refer to [1,3,10,14,15,18,28] and the references therein. Furthermore, the Poincaré-Birkhoff theorem is also an effective tool for studying the existence and multiplicity of periodic solutions of (1.1). It is well known that the global existence of all solutions is generally a basic assumption for the applications of the Poincaré-Birkhoff theorem, even for p = 2 [6,20,21,22,23]. When p = 2, influenced by works in the papers [7,16] and two classical papers [11,12], Qian, Wang and Torres [24] recently developed a new approach to deal with the problems without both global existence of the associated Cauchy problems and the sign assumption on f (t, x). Precisely, they [24] used a rotation number approach and the phase-plane analysis of spiral properties to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems, at zero and infinity. Particularly, [24] obtained multiplicity results with a piecewise linear setting, i.e.
When p = 2, the case is relatively more complex. Recently, by the use of a version of the Poincaré-Birkhoff theorem [5,9,25,22], Yan and Zhang [27] utilized the difference between the rotation numbers of asymptotically p-linear equations of (1.1) at the origin and infinity to find precise multiple 2π p -periodic solutions of (1.1). Precisely, they [27] used the following assumption (f 1 ) there exist ω 0 , v 0 , ω ∞ , v ∞ ∈ L 1 (R/2π p Z) such that the following limits and exist and are uniform a.e. in t ∈ [0, 2π p ]. Assumption (f 1 ) guarantees that all solutions of Cauchy problem associated with (1.1) are defined globally on [0, 2π p ]. Besides the assumption (f 1 ), the uniqueness of the Cauchy problem is assumed in [27]. We refer to [2,17,19] and the references therein for other related papers in the p-Laplacian context, which assume some conditions to guarantee that all solutions of the associated Cauchy problems exist uniquely on the whole t-axis. We also notice that in [24], the uniqueness of solutions for the Cauchy problems associated with the equivalent system of (1.1) (p = 2) is required. In particular, when y = f (t, x) changes the sign at y = 0, in order to describe the spiral property of the solution with a given initial value at y = 0, the authors of [24] used a series of the solutions of analytic systems to approximate the unique solution of the original systems (1.2) with given initial value (see, Lemma 3.1 in [24]). When p > 2, the right-hand side term −φ q (y) of the equivalent system of (1.1) is only Hölder continuous at y = 0. Then, the uniqueness for the associated Cauchy problem is not easy to check.
In this paper we provide an extension to the p-Laplacian case, without assuming uniqueness of solutions, of the results presented in [24] for the case p = 2, and of some results given in [27] for the p-Laplacian but with more restrictive assumptions.
To overcome the lack of uniqueness, we use a recent version of the Poincaré-Birkhoff theorem for Hamiltonian systems by Fonda and Ureña ([8], [7]). We also provide a different method to improve the phase-plane analysis of the spiral properties when y = f (t, x) changes the sign at y = 0.
We will consider a general piecewise p-linear setting, i.e., we assume that The following is the primary result of this paper, where the usual notations, x + = max{x, 0} and x − = max{−x, 0}, are used.
and (H r 0 ). Denote by ρ(a ∞ ) and ρ(b 0 ) the rotation numbers of the piecewise p-linear equations respectively. If ρ(a ∞ ) > ρ(b 0 ), then for any rational number j/m ∈ (ρ(b 0 ), ρ(a ∞ )), equation (1.1) has at least two 2mπ p -periodic solutions. Furthermore, such 2mπ pperiodic solutions have exactly 2j zeros in [t 0 , t 0 + 2mπ p ]. Remark 1. The definition of rotation number will be given in the next section, see also in [27]. Theorem 1.1 improves the corresponding result of [27]. Theorem 1.1 gives also an extension to the p-Laplacian case, without uniqueness assumption, of the results presented in [24] for the case p = 2.
The model example is a partially p-superlinear Laplacian equation, i.e., for (1.1), we assume that where l(t) ∈ L 1 (R/2π p Z) and I is a set of positive measure. We can apply Theorem 1.1 above to prove the following result, which extends the results of [24, Corollary 1.1] and the classical results of Jacobowitz [12] and Hartman [11] to the partially p-superlinear Laplacian equation. The remaining sections of the paper are organized as follows. In Section 2, we introduce the fundamental notion of rotation number in the p-polar coordinates transformation and present certain auxiliary lemmas. In Section 3, we give the phase plane analysis of the spiral properties. Section 4 is devoted to the proofs of Theorem 1.1 and Corollary 1. The Appendix (Section 5) recalls, and further augments, the primary tools for the sake of clarity and coherence.
2. Definition and properties of the rotation numbers. Let p ∈ (1, ∞), denote by q ∈ (1, ∞) the conjugate number of p, that is 1/p + 1/q = 1. Yan and Zhang [27] introduced the following p-polar coordinate transformation of the plane R 2 where (C p (·), S p (·)) is the unique solution of satisfying (x(0), y(0)) = (1, 0). We also refer [29] for the p-polar coordinate transformation of the plane R 2 . Now, we recall the basic properties of C p (θ) and S p (θ) (see [27]), which is beneficial for our purposes.
Let y = −φ p (x ). Equation (1.1) can be rewritten as a planar system In the following, we will write the solution of (2.2) as z(t) = (x(t), y(t)) ∈ R 2 . If z(s) = 0 for every s in interval [t 0 , t], we can define the t-rotation number of z(t) as where θ(t) is the argument function of z(t) associated with the p-polar coordinates (2.1). Indeed, Rot(z(t); [t 0 , t]) describes counter-clockwise rotations performed by z(t) around the origin, in (x, y) phase-plane and the time interval [t 0 , t]. Moreover, for a positive integer j, Rot(z(t); [t 0 , t]) > j(< j) implies that z(t) performs more (less) than j counter-clockwise rotations around the origin in (x, y) phase-plane. In the following, we write by Rot f (t; z) = Rot(z(t); [t 0 , t]) for short. When (2.2) is a piecewise p-linear system the argument function θ(t) satisfies where ω ± (t) ∈ L 1 (R/2π p Z). Thus, A(t, θ; p) is independent of r. By applying Lemma 2.1, we have Then, θ(t) is well defined for all t ∈ [t 0 , t 0 + 2π p ]. Furthermore, A(t, θ; p) is 2π pperiodic in both t and θ. Thus, Eq.(2.4) is a differential equation on a torus. Therefore, we can define the rotation number corresponding to (2.4) as where θ(t; t 0 , θ 0 ) is the unique solution for (2.4) with the initial condition θ(t 0 ; t 0 , θ 0 ) = θ 0 . Yan and Zhang [27] gived further details of ρ(ω).
In analogy to Lemma 4.1 [27], we can prove the following relations for the rotation number of ρ(ω) and the t-rotation number of the solution of (2.3).
Further, we give the following comparison with the result (Lemma 2.3) associated with the t-rotation number of the solution of (2.2). The proof of Lemma 2.3 is given in the Appendix. Hereafter, we write |z(t)| p := |x| p + p|y| q /q for simplicity, where z(t) = (x(t), y(t)).
-Carathéodory function and 2π pperiodic in the first variable, and let ω ± (t) ∈ L 1 (R/2π p Z) be such that where θ 0 is the polar angle of z(t 0 ) in the p-polar coordinates. If where θ 0 is the polar angle of z(t 0 ) in the p-polar coordinates.
Remark 2. Similar to [4], Lemma 2.3 does not require the global continuability of the solutions on [t 0 , t 0 + 2π p ]. The claims of this lemma have to be considered only in regard to those solutions z(t) of (2 Furthermore, we write Rot (2.2) m (z(t)) := Rot(z(t); [t 0 , t 0 + 2mπ p ]), where z(t) is a solution of system (2.2). We can find the following relations of Rot (2.2) m (z(t)) and the rotation number ρ(ω) of (2.3).
As the proofs of the statements are identical, we will only prove the claim (i). By using Lemma 2.2, we have where v = (1, θ 0 ) and θ 0 is the polar angle of z(t 0 ) in the p-polar coordinates. Thus, we can choose a suitable ε > 0 such that 3. The phase plane analysis of the spiral properties. We will prove that the solutions of the system (2.2) have a spiral property described by Lemma 3.1 under (H l ∞ ). Notice that for (2.2), the solution starting from an initial point at y = 0 is not necessarily unique. So, in the part of dealing with the spiral property of the solution from y = 0, we could not follow the approximation method used in [24] which is based on the uniqueness of the solution with given initial point. We will make some improvements of phase-plane analysis of the spiral properties. We will replace the analysis of the solutions starting from a small neighborhood of y = 0 instead of the analysis of the solutions starting from y = 0.
In particular, we will find a priori estimates of a compact set E δ , which can be used to show that large solution of (2.2) from y = 0 only meet {y = ±δ} ∩ E δ finite times. Then, by use of the properties of E δ and energy functions v(x, y) and u(x, y), we can discuss the spiral properties of the solution starting from |y| < δ recursively (see the proof of the Lemma 3.1, Case 3).
Step 1. We first prove that, there exist ξ ± , or there exists t 2 ∈ (0, 2π p ) such that the latter inequality holds for t ∈ [0, t 2 ) and z(t) meet x = 0 entering into D 3 at t = t 2 (see Fig.1). Thereby, we consider the estimates of z(t) as follows: Then then the terms |x| and |y| have at least one, which tends to +∞ as r 0 → +∞. Thus, On the other hand, we consider an energy function v(x, y) = |y| q p + |x| p q .
Note that we are not assuming the sign condition sgn(x)f (t, x) > 0 for sufficiently large |x|. Thus z(t) could eventually return back to D 1 for sufficiently large |x|.
Let t 2 ∈ [t 1 , t 2 ) be a time such that y(t 2 ) = 0. We first consider z(t) in the situation when x(t) < −M ε0 for t ≥ t 1 . We will estimate z(t) the following two cases (see Fig. 1 . We affirm that an estimate of r(t) for t ∈ [t 2 , t * 2 ] only depends on r(t 2 ). Case 2. For case (a), we have z(t) ∈ D 2 and x(t) < −M ε0 , t ∈ [t 2 , t * 2 ). Using (3.4) and the Young inequality, we have , y(t)). (3.10) By using the similar argument for (3.7), we have .
Step 2. Similarly, we can discuss the cases of z(t) ∈ D 3 and D 4 . To conclude, we can find ξ ± N0(4) (r 0 ) such that , for t ∈ [0, 2π p ], or there exists t 1 ∈ [0, 2π p ), such that z(t) intersects x = 0 at t = t 1 and z(t) completes one counter-clockwise turn around the origin when t ∈ [0, t 1 ]. Moreover, Finally, it is clear that ξ ± N0 (r 0 ) can be chosen as strictly increasing functions. For any t 0 ∈ (0, 2π p ], N 0 > 1 and m > 1, the conclusion can be proved by similar arguments.

Modified Hamiltonian systems and the existence of periodic solutions.
Notice that assumptions (H l ∞ ) and (H r 0 ) cannot guarantee that all solutions of Cauchy problems associated with (2.2) are defined globally on [0, 2π p ]. Thus the Poincaré map may not be well-defined. Then, we consider a modified planar system of (2.2). Define a Hamiltonian function where F (t, x) = x 0 f (t, s) ds, K(r 2 ) = K(|x| p + p|y| q /q) ∈ C ∞ (R + , R) is a truncating function satisfying r ≤r 1 ; smooth connection,r 1 < r <r 2 ; 0, r ≥r 2 , wherer 1 ,r 2 are positive parameters, whose specific values will be given in the proof of Theorem 1.1. Thus the associated Hamiltonian system is The solution of (4.1) will also be simply denoted by z(t). It is easy to check the following lemma.
When z(t) does not attain the origin, p-polar coordinates are well-defined.
Lemma 4.2. For any t 2 > t 1 , the nonzero solutions of (4.1) satisfy Proof. If x = 0 and y = 0, then F (t, x) = 0. It follows that x y = −|y| q < 0 for x = 0 and y = 0. Then a nonzero solution of (4.1) performs the counter-clockwise rotations at y-axis in this case. If the trajectory moves from the positive (negative) y-axis to the negative (positive) y-axis, the angular function θ(t) of (x, y) will increment π p . When the trajectory moves back and forth between the second and third (first and fourth) quadrants, the increment of θ(t) will be more than −π p . Thus, for any t 2 > t 1 , we have θ(t 2 ) − θ(t 1 ) > −π p .
To complete the proof of Theorem 1.1, we apply a recent version of Poincaré-Birkhoff theorem for Hamiltonian systems (see [8,7]). Precisely, let 0 < R 1 < R 2 and consider the annulus in R 2 defined as Ω = B R2 \B R1 , where B Ri is an open ball of radius R i , i = 1, 2. Let z(t) be the solution of (4.1), |z(t)| = x 2 (t)+y 2 (t), and define rot m (z(t)) := ϑ(t 0 + 2mπ p ) − ϑ(t 0 ) 2π with the standard polar coordinates Indeed, rot m (z(t)) describes counter-clockwise rotations performed by the solution z(t) of (4.1) around the origin, in (x, y) phase-plane and the time interval [t 0 , t 0 + 2mπ p ]. Moreover, for a positive integer j, rot m (z(t)) > j(< j) implies that z(t) performs more (less) than j counter-clockwise rotations around the origin in (x, y) phase-plane.
is a solution of system (4.1). Then, Rot (4.1) m (z(t)) > j(< j) implies that z(t) performs more (less) than j counter-clockwise rotations around the origin in (x, y) phase-plane. It follows that As a consequence of Theorem 5 in [7], we have Assume moreover that there is a positive integer j such that rot m (z(t)) < j, if |z(t 0 )| = R 1 , and rot m (z(t)) > j, if |z(t 0 )| = R 2 .
Then, the Hamiltonian system has at least 2 distinct 2mπ p -periodic solutions z i (t) with z i (t 0 ) ∈ Ω, such that rot m (z i (t)) = j, i = 1, 2.
Proof. Proof of Theorem 1.1. We will divide the proof into the following four steps.
Then (4.7) holds. Finally, if there exists t 2 ∈ (t 0 , t 0 + 2mπ p ) such that r(t 2 ) > R ∞ , the validity (4.7) is proven by the same arguments as given above.
Furthermore, by using Lemma 4.2, we have which contradicts (4.8). Therefore, |z 1 (t)| p ≤ R ∞ =r 1 , ∀ t ∈ [t 0 , t 0 + 2mπ p ], i.e., z 1 (t) is a 2mπ p -periodic solution of (2.2) with precisely 2j zeros in [t 0 , t 0 + 2mπ p ]. Using a similar argument as above, z 2 (t) is also a 2mπ p -periodic solution of the same node structure. The proof of Theorem 1.1 is thus completed. Now we give the proof of the claim on the partially p-superlinear Laplacian equation in Corollary 1.

5.
Appendix. Here we recall and also introduce, some technical tools that have been used in the proof of the existence and multiplicity of periodic solutions of the p-Laplacian equations considered in the preceding sections.
The following basic properties associated with the piecewise p-linear systems are needed for the proof of Lemma 2.3. As in the proof of Lemma 2.3 in Zhang [29], we can find that t-rotation number Rot ω (t; v) of equation (2.4) depends upon ω ± (t) as follows.