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
Note that if
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
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
Obviously,
For convenience, we introduce some notations
We will also need the function
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.
Theorem 1.1Assume that
(i) If
(ii) If
(iii) If
(iv) If
Corollary 1.2Assume that
(a) If
(b) If
Remark 1.1 The results of Theorem 1.1 and Corollary 1.2 are different with the case
2 Preliminaries
Throughout the paper
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
Lemma 2.1Let
(i) ϕis convex up on
(ii) For each
(iii) For each
Proof By a simple computation, it follows that
(ii) For each
and for each
(iii) By a similar argument, it is not difficult to compute that for each
Lemma 2.2Let
Then
Proof By integrating, it follows that (2.1) has the unique solution given by
where C is such that
Since
Note that from Lemma 2.2, there exists
and for
Lemma 2.3 ([[11], Lemma 4.1 and Lemma 4.2])
For each
whereCis such that
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
(i) If
(ii) If
From Lemma 2.2, problem (1.3) is equivalent to the fixed point problem
in the space E, where
which is a contradiction. This together with Lemma 2.3 implies that
and
Lemma 2.5Let
where
Proof From the definition of
□
Lemma 2.6Let
where
Proof From problem (1.3), since
Let v be the solution of the problem
Then we have
and by a comparison argument, we get 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
Note that
where
which is a contradiction. Thus,
where
where
If
If
Then
where
which is contradiction. Hence
By a similar argument as before, it follows that
where
and subsequently,
Let
□
Lemma 2.7Let
where
Proof Obviously, for any
□
Lemma 2.8Let
where
Proof By using a similar argument of the proof of Lemma 2.6, we have that
If
Let
□
3 Proof of the main results
Proof of Theorem 1.1 (i) Choose a suitable number
where
If
Then Lemma 2.5 implies that
From Lemma 2.4, it follows that
Therefore,
(ii) Choose a suitable number
such that
If
Clearly,
This together with Lemma 2.4 implies
Therefore,
(iii) Since ϕ is a bounded operator, multiplying (1.3) by
Let
with
Choose two numbers
By Lemma 2.8, there exist
and
such that for
This together with Lemma 2.4 implies
Since
On the other hand,
That is,
Therefore,
(iv) Choose two numbers
By Lemma 2.7, there exists
That is,
Since
On the other hand,
Let
That is,
Therefore,
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,
Example 3.2 Let us consider the following problem:
Obviously, we divided the discussion into two cases as follows.
Case 1.
In this case,
Case 2.
In this case,
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
Remark 3.2 Since
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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