Abstract
In this work, we establish the existence and multiplicity results of positive solutions for onedimensional prescribed mean curvature equations. Our approach is based on fixed point index theory for completely continuous operators which leave invariant a suitable cone in a Banach space of continuous functions.
MSC: 34B10, 34B18.
Keywords:
mean curvature equations; positive solutions; existence; fixed point index1 Introduction
The prescribed mean curvature problems like
have attracted much attention in recent years, see [14] and the references therein. Since the problem is quasilinear nonuniformly elliptic, it is more difficult to study the existence of classical solutions. The greatest obstacle is the lack of gradient estimate, such kind of estimate does not hold in general and boundary gradient blowup may occur. This leads to some new phenomena very different from those in semilinear problems. Many wellknown results of semilinear problems have to be reconsidered for this quasilinear problem. Motivated by the search for solutions of the above problem, many authors (see [513]) studied the existence of (positive) solutions for onedimensional prescribed mean curvature equations with Dirichlet boundary conditions
where is a constant, and for and .
Note that if , problem (1.1) is degenerate to the secondorder ordinary differential equation boundary value problems
The existence of (positive) solutions of (1.1) has been well known with various qualitative assumptions of the nonlinearity f, see [14,15] and the references therein.
If , Bonheure et al.[6], Habets and Omari [9], Kusahara and Usami [10], Pan and Xing [12,13] studied the existence of (positive) solutions of (1.2) by using the variational method, lower and upper solutions method and time mapping method, respectively.
However, to the best of our knowledge, the existence and multiplicity of positive solutions for (1.1) are relatively few by the fixed point index theory. In this paper, based on the fixed point index theory, we shall investigate the existence and multiplicity of positive solution of (1.1) when f is ϕsuperlinear and ϕsublinear at 0 and ∞, respectively, here .
Let . Then (1.1) can be rewritten as
Obviously, is an odd, increasing homeomorphism with .
For convenience, we introduce some notations
We will also need the function , and let . By a similar method in [[14], Lemma 2.8], it is not difficult to verify that .
In the rest of this paper, we shall study the existence of positive solutions of (1.3) by using the fixed point index theory to give a brief and clear proof for the existence of positive solutions of (1.1). More concretely, we shall prove the following.
(i) If, then there existssuch that (1.1) has a positive solution for.
(ii) If, then there existssuch that (1.1) has a positive solution for.
(iii) If, then there existssuch that (1.1) has at least two positive solutions for.
(iv) If, then there existssuch that (1.1) has at least two positive solutions for.
Corollary 1.2Assume thatandfor.
(a) Ifand, then there existssuch that (1.1) has a positive solution for.
(b) Ifand, then there existssuch that (1.1) has a positive solution for.
Remark 1.1 The results of Theorem 1.1 and Corollary 1.2 are different with the case which is the classical Dirichlet boundary value problem (1.2). This phenomenon is a striking feature of problem (1.1), which is just the reason why we study the existence of positive solutions of problem (1.1). It is pointed out that in equation of (1.1), having replaced with , Theorem 1.1 and Corollary 1.2 also hold as well as all of the proofs with obvious changes.
2 Preliminaries
Throughout the paper will denote absolute value, and let
Then E is a Banach space endowed with the norm .
We first establish some preliminary results to prove our main result. An easy but useful property of ϕ and is the following one.
Lemma 2.1Let. Thenis an odd, increasing homeomorphism with. Moreover, ϕhas the following properties:
(i) ϕis convex up onandis concave up on.
(ii) For each, there existssuch that, withand for each, there existssuch that, with.
(iii) For each, there existssuch that, . For eachwith, there existssuch that, .
Proof By a simple computation, it follows that and . So ϕ is an odd, increasing homeomorphism with . Moreover, from , we get that ϕ is convex up on . Notice that is also an odd, increasing homeomorphism with . It is easy to verify that is concave up on .
(ii) For each , there exists such that
and for each , there exists such that
(iii) By a similar argument, it is not difficult to compute that for each , there exists such that , . For each with , there exists such that , . □
Lemma 2.2Letwith. Assume thatwis the solution of
Proof By integrating, it follows that (2.1) has the unique solution given by
where C is such that . Hence we must have . Further, since , by using , we obtain , and follows, here .
Since , there exists such that and it follows from that is decreasing on . Then for and for . Hence, on . □
Note that from Lemma 2.2, there exists , , such that with depending on . Define the cone P in E by
Lemma 2.3 ([[11], Lemma 4.1 and Lemma 4.2])
For each, (2.1) has a unique solution given by
whereCis such thatwith. Moreover, the operatoris continuous and sends equicontinuous sets ininto a relatively compact set inE.
We next state the fixed point index theorem which will be used to prove our results.
Lemma 2.4 ([[16], Chapter 6])
LetEbe a Banach space andPbe a cone inE. Assume that Ω is a bounded open subset ofEwith, and letbe a completely continuous operator such that, .
From Lemma 2.2, problem (1.3) is equivalent to the fixed point problem
in the space E, where is such that with , since otherwise,
which is a contradiction. This together with Lemma 2.3 implies that is a completely continuous mapping. Moreover, for any fixed , we have
and . In addition, from Lemma 2.2, it follows that on and there exist , , such that . So is a completely continuous operator.
Lemma 2.5Letbe given. If there existssmall enough withsuch that, then
whereis defined as in Lemma 2.1(iii).
Proof From the definition of , for any , we have
□
Lemma 2.6Letbe given. Ifandfor, then
Proof From problem (1.3), since , it follows that there exists such that . Let . Then u satisfies the following boundary value problem:
Let v be the solution of the problem
Then we have
and by a comparison argument, we get that on . In fact, from , , there exists such that , i.e., . Thus, by a simple computation, we have that
and
This together with ϕ is an increasing homeomorphism implies that
Integrating from 0 to x for (2.6) and integrating from x to for (2.7), respectively, we have that for .
Note that
where is such that , and hence . If , then it follows that
which is a contradiction. Thus, . Moreover, we have
where . Consequently, for . Obviously,
where satisfies . It follows from that . Therefore,
Then
where satisfies , i.e., . If , then
which is contradiction. Hence .
By a similar argument as before, it follows that on . Moreover,
where . So , . Therefore, we have
and subsequently,
□
Lemma 2.7Letbe given. If, then
Proof Obviously, for any , it follows that for . So we have
□
Lemma 2.8Letbe given. If, then
Proof By using a similar argument of the proof of Lemma 2.6, we have that for . Meanwhile, (2.8) is true. If , then we can get that
If , then (2.10) holds and it follows that
□
3 Proof of the main results
Proof of Theorem 1.1 (i) Choose a suitable number . By Lemma 2.8, we have
where
If , then , and so we can choose such that , where small enough satisfies
Then Lemma 2.5 implies that
From Lemma 2.4, it follows that and . By using the additivityexcision property of the fixed point index [16], we have that
Therefore, has a fixed point in . Consequently, (1.1) has a positive solution for .
(ii) Choose a suitable number . By Lemma 2.7, there exists
such that for and . That is, .
If , then there exists such that for , where is chosen large enough so that
Clearly, for , . From Lemma 2.6, we get that
This together with Lemma 2.4 implies . By using the additivityexcision property of the fixed point index [16], we have
Therefore, has a fixed point in . Consequently, (1.1) has a positive solution for .
(iii) Since ϕ is a bounded operator, multiplying (1.3) by and integrating from 0 to , we get that
Let be the solution of . Then there exists such that
By Lemma 2.8, there exist
and
This together with Lemma 2.4 implies , .
Since , from the proof of the case (i), it follows that we can choose such that . Subsequently,
On the other hand, , and (3.3) together with Lemma 2.7 implies
Therefore, has two fixed points and such that and . These are the desired distinct positive solutions of (1.1) for satisfying
(iv) Choose two numbers satisfying
By Lemma 2.7, there exists such that for , we have
Since , from the proof of the case (ii), choose such that . Consequently,
On the other hand, and (3.3) together with Lemma 2.8 implies that
Let and . Then, for any , we have
Therefore, has two fixed points and such that and . These are the desired distinct positive solutions of (1.1) for satisfying (3.4). □
Proof of Corollary 1.2 It is easy to show by the result of Theorem 1.1(i) and (ii). □
Example 3.1 Let us consider the following problem:
Obviously, and . From Theorem 1.1(iv), there exists such that (3.5) has at least two positive solutions for .
Example 3.2 Let us consider the following problem:
Obviously, we divided the discussion into two cases as follows.
In this case, is convex, and and . From Corollary 1.2(a), there exists such that (3.6) has at least one positive solution for .
In this case, is concave, and and . From Theorem 1.1(iv), there exists such that (3.6) has at least two positive solutions for .
Remark 3.1 It is worth to point out that our results only partly generalize the results of Habet and Omari [9] and Pan [12], since is more general than , and due to the limitation of the fixed point index method.
Remark 3.2 Since in (1.1) is autonomous, it is not difficult to show that the solutions of (1.1) are symmetric around , see [2,12]. It follows that in the proof of Theorem 1.1, and we can set , and .
Competing interests
The authors declare that they have no competing interests regarding the publication of this paper.
Authors’ contributions
YL and RM completed the main study, carried out the results of this article and drafted the paper. HG checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
Acknowledgements
The authors are very grateful to the anonymous referee for his valuable suggestions. This work was supported by the NSFC (No. 11361054, No. 11201378), SRFDP (No. 20126203110004), Gansu provincial National Science Foundation of China (No. 1208RJZA258).
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