Positive periodic solution for prescribed mean curvature generalized Liénard equation with a singularity

The main purpose of this paper is to investigate the existence of a positive periodic solution for a prescribed mean curvature generalized Liénard equation with a singularity (weak and strong singularities of attractive type, or weak and strong singularities of repulsive type). Our proof is based on an extension of Mawhin’s continuation theorem.

Inspired by [7,9,17,18], in this paper, we further consider the existence of a positive periodic solution for equation (1.1) by means of an extension of Mawhin's continuation theorem due to Ge and Ren [19]. It is worth mentioning that conditions on f , g and the work for estimating a priori bounds of positive periodic solutions for equation (1.1) are more complex than in [7,9,17,18]. Firstly, the friction term f (u(t))u (t) in [7,9,17,18] satisfies T 0 f (u(t))u (t) dt = 0, which is crucial to estimating a priori bounds of positive periodic solutions for these equations. However, the friction term of this paper, f (t, u(t))u (t), may not satisfy T 0 f (t, u(t))u (t) dt = 0. Secondly, g of this paper possesses weak and strong singularities of attractive type (or weak and strong singularities of repulsive type) at the origin. Thirdly, g of this paper may satisfy sublinearity, semilinearity, or superlinearity conditions at infinity. Therefore, we extend and improve the results in [7,9,17,18].

Positive periodic solution for equation (1.1) when p > 1
In this section, we study the existence of a positive periodic solution to equation (1.1). Since (φ p ( u (t) √ 1+(u (t)) 2 )) is a nonlinear term, coincidence degree theory does not apply directly. The traditional study method is to translate equation (1.1) into the following twodimensional system: , u 2 (t) = -f (t, u 1 (t))u 1 (t)g(u 1 (t)) + e(t), where 1 p + 1 q = 1, for which coincidence degree theory can be applied. However, from the first equation of the above system it is obvious that u 2 < 1, where u 2 := max t∈R |u (t)|. Therefore, estimating an upper bound of u 2 (t) is very complicated; in order to get around this difficulty, we find other methods to study equation (1.1). We first investigate the following second-order prescribed mean curvature equation: Obviously, equation (2.1) can be converted into By [20, Lemmas 3.1 and 3.2], we know that M is a quasilinear operator, N λ is M-compact.
In the following, applying Lemma 2.1, we prove the existence of a positive periodic solution for equation (1.1) with a singularity of repulsive type. Proof We embed equation (1.1) into the following family of equations: where λ ∈ (0, 1]. Firstly, we claim that there exist two points τ , ξ ∈ (0, T) such that Letting t, t ∈ (0, T) be maximum and minimum points of the prescribed mean curvature Applying equations (2.5) into (2.2), we deduce then it is clear that u (t) ≤ 0. By condition (H 2 ) and since g(u(t))e(t) ≤ 0, we get that Similarly, by condition (H 2 ) and equation (2.4), we obtain that u(t) ≤ d 2 . Taking τ = t and ξ = t, (2.3) is proved.
Multiplying both sides of equation (2.2) by u (t) and integrating from 0 to T, we have By condition (H 1 ) and Hölder inequality, the above equality implies 3) and (2.8), using Hölder inequality, we get From equation (2.8) and using Hölder inequality, we deduce On the other hand, let τ ∈ (0, T) be as in equation (2.3). Multiplying both sides of equation (2.2) by u (t) and integrating over the interval [τ , t], where t ∈ [τ , T], we see that Furthermore, from equations (2.2), (2.9) and (2.10), applying Hölder inequality, the above equation implies From equation (2.12), we see that where g + (u) := max{g(u), 0}. Since g + (u(t)) ≥ 0, form conditions (H 2 ) and equation (    In equation (1.2), the nonlinear term g requires a strong singularity of repulsive type (i.e., lim u→0 + 1 u g(ν) dν = +∞). It is clear that the method of Theorem 2.1 is no longer applicable to estimate a lower bound on a periodic solution u(t) of equation (1.1) in the case of a weak singularity of repulsive type (i.e., lim u→0 + 1 u g(ν) dν < +∞). Therefore, we need to find another method to consider equation (1.1) in the case of a weak singularity of repulsive type. Proof We follow the same strategy and notation as in the proof of Theorem 2.1. Next, we consider the lower bound on a periodic solution u(t) of equation (1.1). From equations (2.3) and (2.8), applying Hölder inequality, we get < d 1 . On the other hand, Theorems 2.1 and 2.2 require that g possesses a singularity of repulsive type (i.e., lim u→0 + g(u) = -∞). In the following, we consider that g possesses a singularity of attractive type (i.e., lim u→0 + g(u) = +∞). It is obvious that the attractivity condition and equation (1.2) with (H 2 ) contradict each other. Therefore, we have to find other conditions to consider equation (1.1) with a singularity of attractive type. Proof We follow the same strategy and notation as in the proof of Theorem 2.1. Next, we consider T 0 |g(u(t))| dt. From equations (2.12) and (2.13), we see that where g -(u) := min{g(u), 0}. Since g -(u(t)) ≤ 0, form conditions (H 3 ) and (H 4 ), we know that there exists a positive constant d * 4 with d * 4 > d 3 such that u(t) ≥ d * 4 . Therefore, from equations (2.9) and (2.10), equation (2.16) implies where g -M 1 := max d * 4 ≤u≤M 1 |g -(u)|. The remaining part of the proof is the same as that of Theorem 2.1.
By Theorems 2.2 and 2.3, we obtain the following conclusion.
where μ is a positive constant and μ ≥ 1, n is a positive integer. Proof Let t * , t * ∈ (0, T) be the maximum and minimum points of u(t), and u (t * ) = u (t * ) = 0. Besides, we claim that there exists a positive constant ε such that Assume, by way of contradiction, that inequality (3.1) does not hold. Then u (t) < 0 for t ∈ (t *ε, t * + ε). Therefore, u(t) is strictly decreasing for t ∈ (t *ε, t * + ε), this contradicts the definition of t * . Hence, equation (3.1) is true. Since Applying equations (3.1) into (3.2), we get for t ∈ (t *ε, t * + ε). From equation ( By condition (H 2 ), we get Similarly, by condition (H 2 ), we obtain Therefore, from equations (3.5) and (3.6), we see that By Theorem 2.1, we get that there exist a positive constant M * 2 such that The remaining part of the proof is the same as that of Theorem 2.1.
Comparing Theorems 2.1 and 3.1, Theorem 3.1 is applicable to weak and strong singularities. Theorem 2.1 is only applicable to a strong singularity. However, Theorem 3.1 does not cover the case of p = 2, while Theorem 2.1 covers the case of p = 2. Therefore, Theorem 2.1 can be more general. Besides, Theorem 3.1 requires that g possesses a singularity of repulsive type. In the following, we consider that g possesses a singularity of attractive type. It is obvious that attractivity condition and (H 2 ) contradict each other. By Theorems 2.3 and 3.1, we obtain the following conclusion. It is worth mentioning that the method of Theorem 3.1 is also applicable to the case where g is nonautonomous, i.e., g(u(t)) = g(t, u(t)). Then equation (1.1) is rewritten as the following form: 1 + (u (t)) 2 + f t, u(t) u (t) + g t, u(t) = e(t). (3.8) Applying Lemma 2.1 and Theorem 3.1, we obtain the following conclusion. 1 + (u (t)) 2 + λf t, u(t) u (t) + λg t, u(t) = λe(t), (3.9) where λ ∈ (0, 1). From equation (3.7) and (H 5 ), we get Substituting T 0 (φ p ( u (t) √ 1+(u (t)) 2 )) u (t) dt = 0 into equation (3.11), it is clear that