Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020View this Special Issue
The Existence of Positive Solution of a Nonlinear Problem on Unit Circle
We consider the following nonlinear problem (P) where is a unit circle, is a parameter, and is a nonlinearity. By using the variational method, we obtain the existence of a positive solution to (P) for all . This phenomenon is different from the corresponding boundary value problem.
We consider the following nonlinear problemwhere is a unit circle, is a parameter and the nonlinear term satisfies.
() is a continuous function on , and is odd on .
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 satisfies a superlinear condition as , namely,
uniformly for .
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 , 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 ). If the parameter is small, the related Green function of the operator on 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 on may be sign-changing for some .
In this paper, we aim at studying the existence of a positive solution of nonlinear equation (1) for all and the effect of asymptotic properties of nonlinearity when , 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, satisfiesandis bounded as. Letbe a nonnegative solution of (1). If, thenis 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 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 as the following applications.
Secondly, as an application, we give a strictly positive solution of (1) for all 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.
uniformly for ;
there exists such that, for every and
there exists such that
for every , is an increasing function of on .
Theorem 2. Assume-hold. For, (1) has a positive solution.
We see that the existence of a positive solution in Theorem 2 depends on the assumption that . It is natural to ask whether there exist nontrivial solutions of (1) for some . To shed some light to this problem, we give the existence of positive solutions to (1) with some special nonlinearity as follows.
Theorem 3. Assumewith. Let. Ifis nonnegative with, then (1) has a positive solution.
From Theorems 2 and 3, we obtain that (1) may own a positive solution for all , which is different from the boundary value problem [13–16]. 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 be a solution of (4), we have
If , it follows that there is no positive solution to the boundary value problem (4) with .
Since the proof of Theorem 2 depends on and , 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.
2. 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 is strictly positive by the method of contradiction. Assume for some . We choose . Since , we have two points and in such that and
Since is the minimum value of on , we have
By Taylor’s theorem, we give a formula of for all as follows,
We claim that there exists a strictly monotone decreasing sequence such that
For , we can define a sequence of bywhere . From (11), we see that . Fix , if is large enough, then ; and it follows from (14) that . Following this way, we obtain a strictly monotone decreasing sequence such that and
For each , via applying (10) with , there exists such that
Let in (6), we have
By using the second part of (16) and the assumption that is bounded, we derive that
From (18), we get a contradiction that as .
To prove Theorem 2, we firstly obtain a nonnegative solution of (1) via a minimum of on manifold ; 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.
Lemma 1. (see, for example, Lemma 2.3 in ).Letbe a Banach space,, , , such thatWe denote by the set for any . Then, there exists such that(i), if or if ,(ii)(iii) is an homeomorphism of for any (iv) is non increasing for any
Let be parameterized by angle , be the Sobolev space equipped with the usual normal
Define . Under the assumptions , we have the following variational function of (1).
The related Nehari manifold is defined by
Similar to Lemma 4.1 in , we have the following lemma.
Lemma 2. Assumehold. Letbe nontrivial andfor. If, there existssuch that, . The mapis continuous, anddefines a homeomorphism of the unit sphere ofwith.
Proof. If , we see that for the nontrivial . For any , let . Then is an increasing function of by . By the definition of , we see that , and if and only if , which is equivalent to . Under the assumptions , we see that for small and for large . Therefore, there exists a unique such that =0 and . By , there exists a constant such thatwhere is given by .
Assume that there exists a nontrivial sequence in with , we have a sequence that . This and imply that
It follows that is bounded. And a subsequence of converges to a number . By using (25) and the uniqueness of map , we see that . That is, converge to . The inverse map of is the retraction .
Lemma 3. Assumehold and. Let, then. Ifand, thenis a critical point ofon.
Proof. Let and , where . We prove this Lemma by the following two steps.
Step 1. .
By Lemma 2, we see that if and only if . This and the definitions of imply that . By , we see that for and large. Hence, . From Lemma 2, we see that the Nehari manifold separates into two parts. By assumptions , we deduce that and there exists such thatIt follows that the pass has to cross and . These facts imply that .
Step 2. is a critical point of .
Assume that . Then there exists and such thatFor and , Lemma 1 yields a deformation such that(i) if ,(ii)(iii), We deduce that , which contradicts the definition of .
By Lemma 3, we see that . There exists a sequence of such that as , that is,
Since , we have that is bounded. Then, there exists a weak limitation of the sequence in , i.e., up to a subsequence, weakly in and uniformly on as . This together with the assumptions imply that
It follows from (29) and (31) that , which implies that and . By Lemma 3, we see that . Let and . Then, we deduce that . If is a sign-changing function, then and are nontrivial. It follows that . We thus deduce the following contradiction
So, is not a sign-changing function. By the second part of , we see that . Without loss of generality, we assume that is nonnegative. A standard regularity shows that is on . Since the nonlinear term satisfies assumptions and , we can apply Theorem 1 to get that is strictly positive.
Let us consider the constrained optimal problemwhere
For , it follows from reverse Hölder inequality and (34) thatwhich implies that is bounded in from below under the constrained condition that . By the definition of in (34) and (35), there exists a nonnegative minimizing sequence such that and
It follows from and the Hölder inequality that
Since , by Hölder inequality we see that
This together with imply that is bounded in . So, there is a subsequence such that
By the compactness of Sobolev imbedding, up to a subsequence, we have
It follows that , and therefore .
By the Fatou lemma, we get
So, is a minimizer of under constrained condition that . There exists such that
It follows from the regularity theorem that is a classical nonnegative solution of
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (NSFC) (Nos. 71901222 and 71974204), the Natural Science Foundation of Hubei Province (No. 2019CFB570), and “the Fundamental Research Funds for the Central Universities”, Zhongnan University of Economics and Law (Nos. 2722019PY053 and 2722020JCT008).
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