Periodic solution for prescribed mean curvature Rayleigh equation with a singularity

In this paper, we consider the existence of a periodic solution for a prescribed mean curvature Rayleigh equation with 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.

During the past ten years, the problem of existence of periodic solutions to singular equations has been extensively studied by may researchers [1][2][3][4][5][6][7][8][9][10][11][12][13]. Among these results, some results on Liénard equations with singularity of attractive type (or a singularity of repulsive type) have been published (see [1,5,6,9,10]). For example, Wang and Ma [9] investigated in 2015 a special of equation (1.1), where u (t) √ 1+(u (t)) 2 = u (t) and p = 2, σ = 0, g satisfied a semilinear condition and had a strong singularity of repulsive type, i.e., (1.2) Applying the limit properties of time map, the authors obtained the existence of periodic solution for this equation. After that, Lu et al. [1] improved the results of [9], they required that p > 1. Their proof was based on the topological degree theory. Compared with Rayleigh equations, only a few works focus on prescribed mean curvature Rayleigh equations, especially prescribed mean curvature equations with singularity. As far as we know, prescribed mean curvature 1+(u (t)) 2 of u(t) appears in different geometry and physics [14][15][16][17]. Recently, Li and Ge's work [18] has been performed on the existence of a periodic solution of equation (1.1) with strong singularity of repulsive type by using Manásevich-Mawhin continuation theorem, where p = 2, g satisfied a semilinear condition and equation (1.2). Inspired by the above paper [1,9,18], in this paper, we further consider the existence of a 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 our results are more general than those in [1,9,18]. First, g of this paper satisfies weak and strong singularities of attractive type (or weak and strong singularities of repulsive type) at the origin. Second, g of this paper may satisfy sub-linearity, semi-linearity, and super-linearity conditions at infinity.

Periodic solution for equation (1.1) in the case that p > 1
In this section, we study the existence of a periodic solution to equation (1.1). Since ) is a nonlinear term, coincidence degree theory does not apply directly. The traditional method of studying is to translate equation (1.1) into the following twodimensional system: , u 2 (t) = -f (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, here u 2 := max t∈R |u (t)|. Therefore, estimating an upper bound of u 2 (t) is very complicated; to get around this difficulty, we find other methods to study equation (1.1). We first investigate the following second-order prescribe mean curvature equation: Suppose that the following conditions hold: (i) For each λ ∈ (0, 1), the equation has no solution on ∂Ω.

Corollary 2.1 Assume that conditions (H 1 ), (H 2 ) and equation (1.2) hold. Then equation (1.1) has at least one periodic solution.
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 estimating lower bound of 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 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 of periodic solution u(t) of equation (1.1). From equations (2.3) and (2.8), applying the Hölder inequality, we get  Comparing Theorem 2.1 to 2.2, Theorem 2.2 is applicable to weak as well as strong singularities, whereas Theorem 2.1 is only applicable to a strong singularity. Besides, equation (1.2) is relatively weaker than condition e m-1 m < 2 m-1 αd m-1 1 ω m . On the other hand, Theorems 2.1 and 2.2 require that g satisfies a singularity of repulsive type (i.e., lim u→0 + g(u) = -∞). In the following, we consider that g satisfies a singularity of attractive type (i.e., lim u→0 + g(u) = +∞). It is obvious that attractive condition and equation (1.2), (H 2 ) contradict each other. Therefore, we have to find another conditions to consider equation (1.1) with 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 (2.16) where g -(u) := min{g(u), 0}. Since g -(u(t)) ≤ 0, from 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 The remaining part of the proof is the same as that of Theorem 2.1.
By Theorem 2.3, we obtain the following corollary.

Corollary 2.3 Assume that conditions (H 1 ), (H 3 ), and (H 4 ) hold. Then equation (1.1) has at least one periodic solution.
By Theorems 2.2 and 2.3, we obtain the following conclusion. Finally, we illustrate our results with two numerical examples.
By Theorem 3.1, we get the following corollary. 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, Theorem 2.1 covers the case of p = 2. Therefore, Theorem 2.1 can be more general. Besides, Theorem 3.1 requires that g satisfies a singularity of repulsive type. In the following, we consider that g satisfies a singularity of attractive type. It is obvious that the attractive condition and (H 2 ) contradict each other. By Theorems 2.3 and 3.1, we obtain the following conclusion. By Theorem 3.2, we get the following corollary. It is worth mentioning that the method of Theorem 3.1 is also applicable to the case where g satisfies nonautonomous, i.e., g(u(t)) = g(t, u(t)). Then equation (1.1) is rewritten as the following form:   Finally, we illustrate our results with one numerical example. 1 + (u (t)) 2 + cos 2 t + 3 u (t) 7 + cos 2 t + 1 u 1 5 (t) = sin 2 t + 2 u 3 (t) + e sin 2t , (3.14) where p = 5. It is clear that T = π , f (t, v) = (cos 2 t + 3)v 7 , g(t, u) = -(sin 2 t + 2)u 3 (t) + cos 2 t+1 u 1

Conclusions
In this paper, applying an extension of Mawhin's continuation theorem, we first investigate the existence of a periodic solution for equation (1.1) in the case that p > 1, where g satisfies weak and strong singularities of attractive type or weak and strong singularities of repulsive type, and g may satisfy sub-linearity, semi-linearity, and super-linearity conditions at infinity. After that, we consider the existence of a periodic solution for equation (1.1) in the case that p > 1 and p = 2. Our results are more general than those in [1,9,18].