The Existence of Positive Solution of a Nonlinear Problem on Unit Circle

where S1 is a unit circle, λ is a parameter and the nonlinear term f satisfies. (f1) f ðx, tÞ is a continuous function on S1 ×R, and f ðx, tÞ is odd on t. The second-order differential equation (1) is a nonlinear elliptic equation model arising from studying some physics processes or geometric problems. For example, (1) is the problem of oscillations [1, 2] of a spherical thick shell made of an elastic material when the nonlinearity f satisfies a superlinear condition as t⟶ +∞, namely, ð f2Þ lim t→+∞ f ðx, tÞ/t = +∞ uniformly for x ∈ S 1. For λ = 1, f ðx, tÞ = gðxÞjtjp−1t with p ∈R and gðxÞ: S1 ⟶R, (1) is also used to model the planar Minkowski problem [3–5]. As is well known, the strong maximum principle is a powerful tool for studying positive solutions of a nonlinear elliptic equation, see for example [6, 7]. In [8], the existence of positive solutions to the nonlinear elliptic equation can also be obtained by a standard argument, based on transforming the partial differential equation to the equivalent ordinary differential equation. The iterative method can also be used in the existence of a solution to nonlinear elliptic equation (see [9]). If the parameter λ > 0 is small, the related Green function of the operator d/dx2 + λ on S1 is positive. When the Green function is positive, the upper and lower solution method and Schauder’s fixed point theorem are also suitable to obtain the positive solution to some nonlinear elliptic equations, see for example [10–12]. However, the Green function of operator d/dx2 + λ on S1 may be signchanging for some λ > 0. In this paper, we aim at studying the existence of a positive solution of nonlinear equation (1) for all λ ∈R and the effect of asymptotic properties of nonlinearity f ðx, tÞ when t⟶ 0, to the positivity of solution of (1). Our contribution includes two aspects. Firstly, we establish the following Theorem 1 to show that a nonnegative solution is strictly positive by the analysis based on the Taylor expansion theorem.


Introduction
We consider the following nonlinear problem where S 1 is a unit circle, λ is a parameter and the nonlinear term f satisfies.
(f 1 ) f ðx, tÞ is a continuous function on S 1 × ℝ, and f ðx, tÞ is odd on t.
The second-order differential equation (1) is a nonlinear elliptic equation model arising from studying some physics processes or geometric problems. For example, (1) is the problem of oscillations [1,2] of a spherical thick shell made of an elastic material when the nonlinearity f satisfies a superlinear condition as t ⟶ +∞, namely, ðf 2 Þ lim t→+∞ f ðx, tÞ/t = +∞ uniformly for x ∈ S 1 .
As is well known, the strong maximum principle is a powerful tool for studying positive solutions of a nonlinear elliptic equation, see for example [6,7]. In [8], the existence of positive solutions to the nonlinear elliptic equation can also be obtained by a standard argument, based on trans-forming the partial differential equation to the equivalent ordinary differential equation. The iterative method can also be used in the existence of a solution to nonlinear elliptic equation (see [9]). If the parameter λ > 0 is small, the related Green function of the operator d/dx 2 + λ on S 1 is positive. When the Green function is positive, the upper and lower solution method and Schauder's fixed point theorem are also suitable to obtain the positive solution to some nonlinear elliptic equations, see for example [10][11][12]. However, the Green function of operator d/dx 2 + λ on S 1 may be signchanging for some λ > 0.
In this paper, we aim at studying the existence of a positive solution of nonlinear equation (1) for all λ ∈ ℝ and the effect of asymptotic properties of nonlinearity f ðx, tÞ when t ⟶ 0, to the positivity of solution of (1). Our contribution includes two aspects. Firstly, we establish the following Theorem 1 to show that a nonnegative solution is strictly positive by the analysis based on the Taylor expansion theorem. Theorem 1. Assume λ ∈ ℝ, f satisfies ð f 1 Þ and f ðx, tÞ/t is bounded as t ⟶ 0 . Let v ∈ C 2 ðS 1 Þ be a nonnegative solution of (1). If v≡0 , then v is strictly positive.
Indeed, we can also apply the Green function or the strong maximum principle to study the positive of the solution. However, the direct application of Green function for studying the strictly positive of solution often needs more assumptions on parameter λ and the information of sign of the non-homogeneous term. By using Theorem 1, we could estimate the strictly positive of solution to (1) for all λ ∈ ℝ as the following applications.
Secondly, as an application, we give a strictly positive solution of (1) for all λ ∈ ℝ by using Theorem 1 and the variational method. To study the existence of positive solution to (1) under a general nonlinearity, we also need the following general assumptions.
We see that the existence of a positive solution in Theorem 2 depends on the assumption that λ < 0. It is natural to ask whether there exist nontrivial solutions of (1) for some λ ≥ 0. To shed some light to this problem, we give the existence of positive solutions to (1) with some special nonlinearity as follows.
From Theorems 2 and 3, we obtain that (1) may own a positive solution for all λ ∈ ℝ, which is different from the boundary value problem [13][14][15][16]. To illustrate that the conclusions above for (1) is a new phenomenon that different from the existence of a positive solution to boundary value problem, we consider the following problem Let u ∈ C 2 ð½0, 2πÞ be a solution of (4), we have If λ > 1/4, it follows that there is no positive solution to the boundary value problem (4) with gðxÞ ≥ 0.
Since the proof of Theorem 2 depends on ð f 1 Þ and λ < 0, it is impossible to deduce Theorem 3 by a similar method for Theorem 2. In the next section, we apply a reverse Hölder's inequality and constrained variational method to prove Theorem 3.

Proof of Theorems
We first give our new method to estimate the positivity of the solution of (1) as follows. By assumption, we obtain We will prove that v is strictly positive by the method of contradiction. Assume vðθ 0 Þ = 0 for some θ 0 ∈ ½0, 2π. We choose θ 0 = inf fx ∈ ½0, 2π: vðxÞ = 0g. Since v≡0, we have two points θ r and θ l in ½0, 2π such that By ðf 1 Þ, we see that f ðx, 0Þ = 0. It follows from (6) and (7) that By Taylor's theorem, we give a formula of vðxÞ for all x ∈ ðθ r , θ l Þ as follows, We claim that there exists a strictly monotone decreasing sequence fθ n g ⊂ ðθ r , θ l Þ such that θ n ⟶ n θ r and |v″ θ n ð Þ| > 0: Otherwise, there exists a small δ > 0 such that |v ″ ðxÞ | = 0 for all x ∈ ðθ r , θ r + δÞ. By applying (10), we derive that which contradicts to the second part of (7). By (9), we see that v″ðθ r Þ = 0. Let θ 1 be given by (11), then we have

Journal of Function Spaces
For k ∈ ℕ, we can define a sequence of ρ k by where δ k = ðρ k − θ r Þ/2. From (11), we see that θ n ⟶ n θ r .
Fix k ∈ ℕ, if i ∈ ℕ is large enough, then θ i ∈ ðθ r , ρ k Þ; and it follows from (14) Following this way, we obtain a strictly monotone decreasing sequence fρ n g ⊂ ðθ r , θ l Þ such that ρ n ⟶ n θ r and For each n ∈ ℕ, via applying (10) Let x = ρ n in (6), we have Substituting vðρ n Þ in (17) by using the first part of (16), and then multiplying (17) by 1/v ″ ðρ n Þ, we deduce that By using the second part of (16) and the assumption that f ðθ, tÞ/t is bounded, we derive that From (18), we get a contradiction that 1 = oð1Þ as n ⟶ +∞.
To prove Theorem 2, we firstly obtain a nonnegative solution of (1) via a minimum of I on manifold N; then, we prove that it is strictly positive by using Theorem 1. The process is standard for studying the existence of nonnegative solutions to nonlinear elliptic equations by the method involving the Nehari manifold. For completeness, we give the main steps. The following deformation lemma is needed. [17]). Let X be a Banach space, I ∈ C 1 ðX, ℝÞ, S ⊂ X, c ∈ ℝ, ε, δ > 0 such that

Lemma 1. (see, for example, Lemma 2.3 in
We denote by I d the set fu ∈ X : IðuÞ ≤ dg for any d ∈ ℝ. Then, there exists η ∈ Cð½0, 1 × X, XÞ such that ·Þ is an homeomorphism of X for any t ∈ ½0, 1 (iv) Iðηð·, uÞÞ is non increasing for any u ∈ X Let S be parameterized by angle x, H ≔ H 1 ðSÞ be the Sobolev space equipped with the usual normal Define Fðx, uÞ = Ð u 0 f ðx, tÞdt. Under the assumptions ð f 1 Þ − ð f 3 Þ, we have the following C 1 variational function of (1).
The related Nehari manifold is defined by Similar to Lemma 4.1 in [17], we have the following lemma.

Lemma 2.
Assume ð f 1 Þ − ð f 6 Þ hold. Let u ∈ H be nontrivial and hðtÞ = IðtuÞ for t > 0 . If λ < 0 , there exists t u > 0 such that t u u ∈ N, hðt u Þ = max t>0 hðtÞ . The map u ⟶ t u is continuous, and u ⟶ t u u defines a homeomorphism of the unit sphere of H with N.
Proof. If λ < 0, we see that 1/2 Ð S 1 u′ 2 dx − λ/2 Ð S 1 u 2 dx > 0 for the nontrivial u ∈ H. For any u ∈ N, let gðtÞ = 1/t Ð S 1 f ðx, tuÞ dx. Then g is an increasing function of t by ð f 6 Þ. By the definition of hðtÞ = IðtuÞ, we see that hð0Þ = 0, and t u u ∈ N if and only if Journal of Function Spaces we see that hðtÞ > 0 for small t > 0 and hðtÞ < 0 for large t. Therefore, there exists a unique t u such that h ′ ðt u Þ=0 and t u u ∈ N. By ðf 4 Þ, there exists a constant C 0 > 0 such that where α > 2 is given by ð f 4 Þ.
Assume that there exists a nontrivial sequence u n ⟶ n u in H 1 ðS 1 Þ with u=0, we have a sequence ft u n g that t u n u n ∈ N. This and ðf 4 Þ imply that It follows that ft u n g is bounded. And a subsequence of ft u n g converges to a number t 0 . By using (25) and the uniqueness of map t u = tðuÞ, we see that t 0 = t u . That is, t u n converge to t u . The inverse map of u ⟶ t u u is the retraction u ⟶ u/∥u∥. By using the properties of I in Lemma 2. We obtain the following Lemma 3. where Γ ≔ fγ ∈ Cð½0, 1, HÞ: Iðγð1ÞÞ < 0, γð0Þ = 0g. We prove this Lemma by the following two steps.
By Lemma 2, we see that Iðt u uÞ = max It follows that the pass γ ∈ Γ has to cross N and c ≥ inf ∥u∥=r IðuÞ > 0. These facts imply that c 1 ≤ c.
Step 2. v is a critical point of I.
By Lemma 3, we see that c 1 = inf u∈N IðuÞ > 0. There exists a sequence of fu n g ⊂ N such that Iðu n Þ ⟶ c 1 as n ⟶ +∞, that is, Let α > 2 be given by ð f 4 Þ, it follows from (28) and (29) that Since λ < 0, we have that f∥u n ∥g is bounded. Then, there exists a weak limitation v of the sequence fu n g in H, i.e., up to a subsequence, u n ⇀ v weakly in H and u n ⟶ v uniformly on S 1 as n ⟶ +∞. This together with the assump- It follows from (29) If v is a sign-changing function, then v − and v + are nontrivial. It follows that v − , v + ∈ N. We thus deduce the following contradiction So, v is not a sign-changing function. By the second part of ð f 1 Þ, we see that IðvÞ = Ið−vÞ. Without loss of generality, we assume that v is nonnegative. A standard regularity 4 Journal of Function Spaces shows that v is C 2 on S 1 . Since the nonlinear term f satisfies assumptions ðf 1 Þ and ð f 3 Þ, we can apply Theorem 1 to get that v is strictly positive. Let us consider the constrained optimal problem where For λ ≥ 0, it follows from reverse Hölder inequality and (34) that which implies that IðuÞ is bounded in H 1 ðSÞ from below under the constrained condition that FðuÞ = 1. By the definition of I, F in (34) and (35), there exists a nonnegative minimizing sequence fu n g ⊂ H 1 ðSÞ such that Fðu n Þ = 1 and It follows from 2/ðp − 1Þ > ðp − 1Þ/ðp + 1Þ and the Hölder inequality that